First things first, your post has many many issues to take in but here's the start, very blunt becuase there is much to cover.
The cheating problem: Do not do this. Any new fling regardless of what woman or how exciting for you at first, will eventually settle into this same routine with you and your wife now because the problem is not your wife, the problem is with you.
Also do not fall into the trap of thinking just if your woman is not interested in sex with you then she is not interested in sex with some affair man.
A woman is needing sexual drive as strong as a mans, and her fire is not lit by you then it is like dry kindling waiting to be lit by maybe some affair man that does light her fire, much like you are tempted to do yourself with some affair woman.
This scenario is far to common and if you are having these issues it is danger to your marriage.
The stay at home dad problem:
I am not known in this forum to be politically correct, so neither in this area as well. Do not underestimate the effects of being a follower to your woman to kill her sexual desire for you, and that is what your post is telling me.
You are needing to stop this following your woman, it is killing any sexual attraction she would have for you.
If you are to stay at home dad that is up to you, but it needs to be YOUR arena that you are running that show and the house duties, not trying to please your woman or other such things, trying to appease your woman or follow her leading.
You are only doing one think in her eyes, looking like a little boy that she is the mother to come behind him to make it proper.
Also this is the same for stay at home dad, as it is for the man who is losing his job or unemployed or not as professional as his woman in making money or any other reason is waiting on his woman to support him, and that is simply this: His attitude is not dominant in such scenarios.
Understand this one thing, no matter what the relationship, the money, the profession, the employment situation, in the sexual relationship one thing matters, that the woman sees her man as in control of himself and his environment, this is both attitude and behavior.
If the man is unemployed and staying at home that is fine, but in attitude he is still to give the behavior and confidence of being in control of himself, whether to work towards education or establish career or even the house husband, regardless he needs to take the ownership of his arena and show his woman the mettle he is made of.
So instead to say 60/40, instead say "This is my responsibility I am taking care of it". So to claim 100 percent of the house duties, even if this means you are delegating some of the tasks to the children, an outside cleaning service, or even to delegate this or that to your wife. See though the difference, that is ownership, and that is dominance, and that will make you look very much a man in your woman's eyes, and not the little boy!
And to this worrying about upper body lifting? Look, if this torn rotator cuff is causing this kind stress to you, you need to count your blessing because there is much worse things to be happening then 9 months of upper body atrophy.
Weight injuries are a your body commnicating to you, either there is problem with form or nutrition or rest, but listen to your body. Instead to worry about this or that upper body, instead to enjoy the "downtime" and do, maybe, even more squats and lower and full body exercises.
And after the injury is healed, keeping up with squats and lower body work, your upper body routine will return with a vengence!
I wish you well.