we own cats for rodent control, one of which was a stray that managed to win us over. The major problem I have with strays, is the disease problem. ( the damage to small stock is, where I am , nearly unavoidable to some degree, and a sign to build better enclosures)
For most of the dumped animals, putting them down is a kindness, at least it is over quickly.
Keep your own animals well looked after ( including shots), and they will help move on strays, but at least sometimes the answer is to trap them if possible and either have them picked up by some animal control agency, or put them down as painlessly as possible.
FIV and extreme infestations of worms and fleas give very very nasty ways for these animals to go, even assuming that they are able to catch enough food on their own