Your question reminds me that children are occasionally taught techniques using open tegatena so as to specifically avoid grabbing. This is done for the purpose of teaching the shape of the movement, rather than the technique itself, since among other reasons this reduces the technique to a more basic level without interference from a poor posture by uke, and reduces chances of over- or under-performance of a movement which might injure.
But your question also reminds me that arthritis - which I suffer horribly from, albeit in the hips - is but one of a myriad of things that can go wrong. One may have a sickness, or injury, or a hand that is occupied (eg, holding a child or a weapon). In these situations, one must take care to understand the underlying focus of the technique and then adapt when the optimal body part used to effect the technique is not able to perform its function. Hands and fingers can become jammed, smashed, cut, burnt, or over extended during any altercation. Maybe even you have a weapon, are you going to give that up to effect a grab?
Your instruction would be incomplete without coming up with scenarios that a student might encounter - injury typically, but an ailment like arthritis is just as good. What does a one-armed student do? Or a student without a hand? Or Leg? As such, this is an excellent question to bring up.
There is no easy answer, since no one answer covers for all scenarios, but you could make the argument that the opponent's own momentum should dictate how they are handled.
So, using your example, some throws which you might have done by sweeps and grabs could be effected by lowering to the knees, forcing the advancing opponent to trip over you. In fact, that lowering of your body by kneeling can often be substituted for many grabs and throws. As you are an instructor, I trust that you could easily come up with many scenarios which you could swap out the grab for a kneel, a spin, a duck, a step here or there. Where you might have issued an atemi to slow the advancing opponent, you omit it in order to keep them moving so that they entrap themselves.
This won't easily work all the time, and you might have to break rules here and there. Perhaps, one hand is no good, but other is fine, so maybe issue a technique allowing a shift to the other side might work.
An example of breaking a rule: instead of stepping away from an opponent, you simply turn the torso to the direction of the would-be step. Try this when you have no functional issues, and your instructor is liable to spend time making corrections about stepping. But if a student cannot turn and step, and turning the torso is all that can be done, then that's what will have to do, even at the cost of having to muscle through a technique.
As it is, I am myself exploring how to adapt with occasionally debilitating arthritis. No more kneewalking or suwari waza for me. And my taekwondo kicking days are all but over. So I am curious as to what other answers you may get.