I think we should have random minions given out...

Like this:

"Minion name here" just followed "pet name here" home.

OR

You tripped over a "Minion name here" guess you better take care of it.

Stuff like that maybe????

I'm pretty sure this is already in place, but just doesn't happen that often. I got a random event the other day with something along the lines of "Aww! This little guy has followed you home" and then had a Bing in my inventory that wasn't there before.

Yeah I've also gotten a minion from a random event.

We also have the minion feeding game that can give randomly give you the minion you feed

You pretty much just described minion feeding.