I too have worked with cats in a shelter situation, it is important that animals and cats in this situation are rehomed but only specific cats should be introduced to the elderly, to give a young cat to this group would be folly. Whereas they would have excellent care etc - if the elderly owners succumb to old age then the cats find themselves in exactly the same situation as before.
I cannot see old folks homes takiong on animals due to the complax issue of running an elderly home, however, in theory you have a point because an old folks home could have a constant or two - a cat or a dog then old folks come and go but the animals remain, as this happens. It would not take care of the majority of lost animals but it could help.
I do agree with you. There are things to be considered first I guess before doing that. Since you want elderly to possess them then you should be aware of the things that you might do in order to pursue that. Assess 1st the behavior of the cats and if you think it would be a burden then don't belong that to the group, instead, do something to change the behavior so the next time it'll be ready.