1x2s should be fine in an urban setting where you won't have large predators that want to break in.
With small dimension lumber like that it's all about triangulation to give more strength (or L-brackets, etc), the pic you posted has a decent amount actually. You'll also want to make sure to pre-drill holes and pay attention to wood splitting since there's not much wood to keep it together. If you're sandwiching 1x2s around HC, that's a good idea for panels and would add more strength since they're doubled up.
I have built many things out of 1x2s and actually have a greenhouse built by the PO of my house that resembles what you posted - that structure was painted wood and it's starting to fall apart, but I want to say it's at least 15yrs old. I have used a lot of 1x lumber in my gardening projects (like small lean-to's, work tables, etc) and those I don't paint, for me in CA they seem to last several years before they start to fall apart.
If I was building a coop out of 1x2s, I'd likely go get a gallon of sealer of choice (I'd probably go water based spar urethane), lay them all out on some saw horses and paint them with it before I cut them.
I know lumber has come down a lot in price lately but a 1x2 is still not the most cost effective lumber choice. Convenient yes. Personally if it's going to be something I want long-term I'd likely get better wood. Heck, even 2x2s.