Looks like a mayfly. You're correct. They don't do anything nasty.

That might very well be it but these don't have any "tails" and I can't imagine where they would have come from. There's no water in the greenhouse.

Well, it's Like this. The tail may be obscured in the curve of the wing.

I found a mayfly at my house last week, sitting on the patio door. And we don't have any water either.

The tail is definitely not obscured, it's not there. I'm not trying to be argumentative and agree that it most resembles a mayfly but it's also a bit late in the year and it's not like it was just a single fly that wandered in. At least 100 all appeared at once so it seems that they would have just emerged. Is there another fly, similar to a mayfly, that doesn't pupate in water?

Reminds me of one of the species of midge. Most of them require semi-aquatic locations in which to pupate but once air-borne, they can be found anywhere. I've seen hoards of them sitting on the undersides of leaves, just like yours.