If farmers have heart of the wild trait, giving them the focused study feat and so choose handle animal, would they be able to use it on creatures such as Stirges, while being magical beasts have an INT of 1, and so make them part of their farm?
Think Ravenmoor.
Do you mean the Heart of the Wilderness human alt racial? If so, that's for survival checks.
Also, there's nothing stopping you from capturing a male and female stirge and penning them in an enclosed/netted pasture, and breeding them. That doesn't take any checks other than the ones necessary to net them and capture them alive. You can use Handle Animal to "rear a wild animal" on the offspring, though. And then you could teach it the "Air Support" or "Guard" tricks. But your Handle Animal skill would be essentially worthless on the original male/female Stirges, as these are wild creatures and cannot be domesticated.
If you're the DM, you could consider using 3pp class Monster Trainer as well.
I am struggling to see how, by RAW, you train Magical Beasts with an Int of 1-2 using just Handle Animal.
You can use Handle Animal to "rear a wild animal" on magical beasts, such as rearing Owlbears from birth. This is GM discretion to allow/disallow though.