This avoids a lot of hassle with the store’s repair team and is the same price as most repairs. (The front glass only repair is slightly cheaper than ERS but wouldn’t you rather get a new phone, battery, etc for slightly more?)
I'm sure plenty fall for it. All those nudes don't leak themselves.
Doesn't answer your question, but be careful that your distrust doesn't lead you into worse trouble.
Unfortunately this doesn’t help your situation, so I’d recommend taking a full offline backup using iTunes, erasing the phone and restoring your backup once you get the device back.
As far as the Apple support app, I think that Apple does have a legitimate app under that name. However, usually find my is disabled under the find my app, not the Apple support app (unless Apple stores use a custom version with it in there or it’s hidden in the regular app where I can’t find it). It’s also possible the support app just redirected you to the find my app. Either way, if you don’t trust the app, you could just go to a web browser at home and disable it via the web app for find my so they can proceed with the repair. I don’t know if third party repair shops will also make you disable find my, but I know Apple usually asks you to disable find my first
Maybe the apple support app uses a similar mechanism to disable find my?
So I just went to a nearby repair guy who only asked for my phone number to update me on the repair status.
I manage RMA for consumer electronics, and we don't do any shit like that.
We just have a factory test firmware that is signed by us, that can be booted from a USB storage, boots in RAM (so it doesn't need anything from local storage except first stage bootloader, and we disable secure enclave in first stage boot in that mode).
It's still a bit of a security tradeoff since that gives attacker an extra surface area to execute code on Application Processor, but still nowhere near giving your password to strangers?
It's common to ask for the password and I don't think they'd touch your private files.
After the screen is fixed they'd need to do QA that it works correctly. Some things you just have to trust in life, too much paranoia won't lead you to good places. Trust me, I've been there.