then pressing power button
says no. If it were shorted you'd be in something like a boot loop with a delay. If it were open, nothing you did (pressing it) would have any effect in any mode. Broken switches don't "fix" themselves when you're booted into a different partition (which is what you're doing).
If the switch is working (it is) and it doesn't turn the power on, it's a firmware problem.
If doing a few things, but NOT touching the power button, made the phone come on in some mode, it would possibly be an open power switch.
It really makes very little difference, though, in your next move. Replacing the power switch puts you in a position in which you can damge a few other things. Trying to fix them opens you to damaging other things. By the time you admit defeat, you either toss what was a perfectly good phone, except for a 69 cent switch, or you bring it to a repair shop, they give you a repair estimate and, after your heart attack (it'll probavly end up costing more than an outright buy of a new phone), you'll still toss the phone.
Bring it in now and pay a lot less to get it fixed right the first time. There are repairs you can do yourself, there are repairs that require years of experience. (Soldering to plastic substrate is one of those that requires experience. Plastic melts at MUCH lower temperatures than solder.) If you cut yourself shaving, you patch it up. If you break a leg you go to a doctor. Same principle applies here - this isn't a shaving nick, it's a broken small bone.
