I think it's generally accepted that sterilisation isn't very effective specifically due to their breeding rates as you describe.

Australia for example has taken to extreme measures of just mass cat genocide ('culling') because there really is no other way.

Not that I want to rain on your parade but, for example here in China, I see cat food being laid out by old ladies and such all over the place, almost every street. They are encouraging cats to breed with mis-directed kindness.

I suspect a more powerful approach would be a societal campaign, spreading the knowledge not to do this, not to feed them when they meow and beg. Teach them that this only leads to increased suffering through having more babies which then don't have the food to eat, and starve (after breeding themselves).

So many people are kind on a very basic level without considering the consequences, which, as we see here in China, is appalling and tragic.