I don't know if anybody noticed this, but if you drop the villager on a hut in a certain way, they will almost always start fixing (as opposed to just examining).

I usually drop them in the above right hand corner of the hut.

It this coincidence or am I really on to something?

I just did a very quick, and probably unrepresentative, test by dropping two different villagers as you described 25 times. They started fixing the hut 18 times and examining it 7 times. Of course, I may not have been dropping the villagers in the exact same spot as you.

Aha, that sounds promising.

For clarity, I drop them on the rooftop, just above the window.