IMO, this circumstance is sufficiently covered by the vacation facility. If you have a serious health problem that's going to take longer than a month or two to resolve, then you have bigger things to worry about than having to lose some games by timeout. Ratings can always be made up after you recover.
In the meantime, as you point out later (contradicting your own point?), the abuse would be ridiculous. I would not enjoy having my opponents beg for an abort after I've spent months achieving a winning position against them.
I think there should be an abort game facilty in addition to agree draw and resign.
Someone has 30 games on the go, contracts a serious illness and has to go into hospital for an unknown length of time. They have more serious things to worry about than chess.com, but want to wrap things up before they go into hospital. They are ahead in 15 of their games, drawing in 10 and losing in 5. They resign the 5 and agree draws in the 10, but what of the 15 they were ahead? Their opponents realise they were ahead in the game, take compassion and agree to abort the game. Maybe 2 or 3 thought the game was drawn, so refuse to abort, but agree to a draw.
The aborted games mean there is no ratings change.
Someone abuses the system by feeding a sob-story to their opponent when they are clearly losing. Opponent can refuse to abort, accepting only a resignation or draw.