There is a Greek word that covers much of this.
2323 therapeúō – properly, heal, reversing a physical condition to restore a person having an illness (disease, infirmity).

[2323 (therapeúō), the root of “therapy” and “therapeutic,” which usually involves natural elements in the process of healing but the use of the greek word therapeúō in the gospels and acts have little to do with natural remedies or cures

https://biblehub.com/greek/2323.htm

— Healing on the Sabbath: A big deal to the Pharisees

The word here is again therapeúō. It is not iáomai which we shall cover next week. We see in Matthew 12 Yeshua asking the question about a sheep falling into a pit on the Sabbath and whether or not any of them would pull out the poor animal that day. This is part of his teaching about what is permitted to do on Sabbath. In no way was Yeshua promoting the idea of working on the sabbath but rather promoting the idea of mercy and doing what is necessary on the sabbath.

Matthew 12:10; behold, there was a man who had a withered hand. And they asked Him, saying, “Is it lawful to heal on the Sabbath?”—that they might accuse Him.

This got recorded again in Mark 3:1 and Luke 6:7 where it was the scribes and Pharisees watching. Another sabbath event happened in…

Luke 13:14;  the ruler of the synagogue answered with indignation, because Jesus had healed (a woman who could not straighten up) on the Sabbath – he called it an infirmity (astheneia) which I will dive into deeper in a few minutes.

In this case the lawyers teamed up with the Pharisees to call it work. Yeshua again retorts with the idea of giving on ox or a donkey something to drink on the sabbath, calling them hypocrites, for they found convenient yet necessary exceptions to the work rule.

Luke 14:3; Jesus, answering, spoke to the lawyers and Pharisees, saying, “Is it lawful to heal on the Sabbath?”

Here he healed a man of dropsy which is swelling in the body or skin. Then there is the criticism called against a man healed of paralysis.

John 5:10; he Jews therefore said to him who was cured (paralyzed 38 years), “It is the Sabbath; it is not lawful for you to carry your bed.”

For near to the pool of Bathesda lay a great multitude of sick people (astheneō), blind (typhlos), lame (chōlos), paralyzed (xēros) waiting for the moving of the water (John 5:3)

So if healing one with a withered hand, dropsy, a back issue, or paralysis was not ok on the sabbath we have to better understand the idea of working on the sabbath. The Pharisees had a very strict interpretation of the law, maybe somewhat like the Ultra Orthodox today. The Pharisees seemed to have the scribes, lawyers, and synagogue rules on their side. Yet Yeshua prioritized mercy over law.

Today pikuach nefesh (saving of a life) rules varies between jewish groups. Most would agree that something critical like a doctor saving a life, a fireman rescuing someone from a buring building, or policeman working to save lives are acceptable work on Shabbat. Other things are less clear and some even interpret driving a car as work, especially when it is unnecessary.

Yeshua also said to the Pharisees ‘you tithe mint and dill and cumin, and have neglected the weightier matters of the law: justice and mercy and faithfulness’. Healing is another instance (in this case mercy) where there are priorities in the law. Yeshua then had a rather liberal interpretation of sabbath (halakha) laws, but that does not mean he intentionally ignored the law to rest on Shabbat. It seems to me that all can pray on shabbat. The real issue is if something like laying on of hands is considered work. This is where the Pharisees started making up legalistic rules for do’s and don’ts.

We need to ensure our actions please God. In Mark 2:27 we hear that Yeshua said ‘the Sabbath was made for man, not man for the Sabbath. Therefore the Son of Man is also Lord of the Sabbath’. Yet a day of rest is a gift from God which we should not ignore.

Let’s go a bit deeper into what Yeshua and his disciples did that seemed so wrong on the sabbath.

— The Sick and Miserably Ill (asthenēs and astheneia)

Remember the sabbath healing of a woman who could not straighten up due to back issues. It was when the 70 were appointed to go heal the since Yeshua told them to heal (therapeuō) the sick (asthenēs). Luke 10:9

Here we have the words therapeuō and asthenēs. The first word is the action and the second the state of the person.

asthenēs (as-then-ace’) = weak, infirm, feeble (ajective)
astheneia (as-then’-i-ah) = without strength, weak, sick (noun)

Not being able to stand up straight could have been a spinal injury or degeneration.

Luke 5:15; great multitudes came together to hear, and to be healed by Him of their infirmities (astheneia)

Luke 8:2; certain women who had been healed of evil spirits and infirmities (astheneia)—Mary called Magdalene, out of whom had come seven demons

I just want to point out that at least three prominent women mentioned here who were healed provided funds for Yeshua’s ministry. Just pointing out some things are more valuable than gold or silver.

Acts 5:16, etc.;  a multitude gathered from the surrounding cities to Jerusalem, bringing sick (asthenēs) people and those who were tormented by unclean spirits, and they were all healed

Technically we cast out unclean demons and heal the sick but here the two concepts are linked together. This is possible. Sometimes we need to cast out before healing. Fear, doubt, and insecurity often block a miracle.

— Infirmities and Afflictions (arrōstos and mastix)

In Nazareth we read that Yeshua did very little miracle work:
Mark 6:5; He could do no mighty work (dynamis) there, except that He laid His hands on a few sick (arrōstos) people and healed them.

In this case, as often, familiarity breeds contempt. They had little faith in him around his home town.

Mark 3:10; He healed many (of ??), so that as many as had afflictions (mastix) pressed about Him to touch Him

mastix = plague, whip, or scourge

Luke 7:21; that very hour He cured (therepeo) many of infirmities (nosos), afflictions (mastix), and evil spirits; and to many blind He gave sight. Yeshua answered the disciples of John say ‘Go and tell John the things you have seen and heard: that the blind see, the lame walk, the lepers are cleansed, the deaf hear, the dead are raised, the poor have the gospel preached to them’.

I don’t have a specific section for the lame and blind nor even those suffering from leprosy.

Acts 4:14;  seeing the man who had been healed (lame from birth) standing with them, they could say nothing against it.

So I would think those things are also infirmities however the leapers are always cleansed.

— Sickness and Disease (nosos and malakia)

Matthew 4:23 (and Matthew 9:35); Jesus went about all Galilee, teaching in their synagogues, preaching the gospel of the kingdom, and healing all kinds of sickness (nosos) and all kinds of disease (malakia) among the people.

Matthew 10:1; when He had called His twelve disciples to Him, He gave them power over unclean spirits, to cast them out, and to heal all kinds of sickness (nosos) and all kinds of disease (malakia).

Mark 3:15; He might send them out to preach, and to have power to heal sicknesses (nosos) and to cast out demons — note that only a few versions like KJV speak of healing sickness (nosos) here

Luke 9:1; He called His twelve disciples together and gave them power and authority over all demons, and to cure diseases (nosos).

The last two verses speak only of healing sickness, not disease. Perphaps that is because demons tend to mask or prevent curing of sickness, not necessarily disease. I would need much more study and insight to say that defintively. Yet I do believe there is a difference between sickness developed because of spiritual oppression and disease which is a merely a physical or heredity condition. I also say this because therapeúō is related to therapies which heal of mental conditions not just physical. The devil mainly impacts us in our minds.

Ex. 15:26 speaks of the diseases (plagues) put upon the Egyptians and corresponds to ‘nosos’ so maybe this word has to do with external things that affect us including demonic activity. In the end the death angel brought death of the firstborn so affects of sickness can be extreme.

— Miserable Ill People (kakōs) and the Tormented (basanizō)

Matthew 4:24; His fame went throughout all Syria; and they brought to Him all sick (kakos) people who were afflicted with various diseases (nosos) and torments (basanos), and those who were demon-possessed, epileptics, and paralytics; and He healed them.

Notice here to that disease (nosos) is related to torments, demon-possession, and mental/motor control.

kakōs = miserable, ill which like asthenēs is descriptive.

Mark 1:34; He healed (therapeuō) many who were sick (kakōs) with various diseases (nosos), and cast out many demons;

Then we read about Yeshua merely speaking the word about a centurions servant who was at a gentile home:
Luke 7:2, a certain centurion’s servant, who was dear to him, was sick (kakōs) and ready to die.

In Matthew 8:5-7 when Jesus had entered Capernaum, a centurion came to Him, pleading with Him, saying, “Lord, my servant is lying at home paralyzed (paralytikos), dreadfully tormented (basanizō).” Yeshua said to the centurion “I will come and heal him.” (a paralyzed man in his house).

This could have been the same servant of the centurion. But the main point here is that going into a gentile’s home was as bad or worse as healing on the sabbath. Yet the man was greatly tormented (mentally and physically). Compassion overrulled here not entering a gentile’s home but I think Yeshua did not need to go in because he merely spoke the word. Still, it was very ackward for Yeshua to go into the gentile home, even though he would have. Again, compassion overrules legalism…

I hope this gives you a new or unique perspective of therapeúō. Next week we shall speak of another healing concept called iaomai in greek. Let me say my going in assumption now is that iaomi is the deals more with miracles and that of a higher degree.

References:
https://fopchurch.net/blog/2020/09/28/greek-week-healing
https://mcmworshipcenter.online/sermons/the-three-types-of-healing-mentioned-in-the-bible/