Correctly measure L7805ACT regulator output
I suspected that something as simple as this had to have a simple (yet improbable) explanation and it did.

Turns out both regulators were busted. I don't remember where I got them from, they looked unused, but who knows. I tried with the same model but bought at a different time, maybe from another supplier, and works as expected.

The testing setup was correct and filter capacitors aren't really needed (at least for testing purposes), but I'll use them anyway as recommended. Thank you all for your input.