"but I feel incredibly limited in what I can actually do with my iPhone"

That's because they're intended to work as media consumption devices rather than productivity ones. If someone feels productive on a phone, he or she has very likely adapted herself to the device than the other way around.

I had to struggle yesterday with my GF new phone because her lawyer sent her a paper in .doc format using Whatsapp (no kidding[1]) and she had no capable reader. Fine, I thought, let's fire up the app store and find one. Now I just have a couple old tablets I keep at home 24/7 and don't use smartphones at all: no feature in the world can convince me an € 5000 smart phone can be more usable than my €20 dumb one plus my € 400 used Thinkpad, so I'm not exposed to the level of annoyances the usual phone user experiences every day.
Back to ther phone, I knew the google app store wasn't that wonderful land where you tap the equivalent of apt-get install name-of-a-free-as-in-unicorns-and-fairies-doc-redaer and it magically installs, but man... I didn't expect that level of crap! I tried a couple "free" reader apps and every attempt to run them had either popups appearing asking to click to buy something, or more subtle ones where the click to buy button was a lot bigger and dangerously close to the window really small X (close) button.
In the end I installed f-droid, dns-66 and libreoffice reader, but what if she had to do all for herself and could only find those pieces of crapware?

[1] apparently that practice is widespread among non IT professionals, and I find it rather dangerous. The privacy implications should raise some big alerts.

"I would absolutely love a phone that could let me run the "real" versions of apps when I need to. "

That's my dream as well. Give us an open (hardware) phone and OS plus apps will follow.
Manufacturers however would hate this because the phone (tablet, etc.) as a platform allowed them to undo what Open Source achieved in the last decades on personal computers: now they have again a piece of hardware they can fully control; the software is closed and incredibly dumb, and it forces you to connect to online services to do anything. We're sort of back to mainframes, that's like killing over 40 years of IT development!. Also by keeping some parts of the firmware (mainly drivers) closed they can ensure no FOSS hippy is going to ruin their dominance. If there was a way to install a working native mainline Linux image (no chroots/VMs/frankendroids etc.) with full hardware support all those phone left in a drawer collecting dust could magically become useable again, with some important side effects.
For starters, I'd dare to say every 10 of them at least three new phones would remain unsold, then it would become a lot harder to put spyware into user phones, and third, what would happen when even a few thousand users in the world started showing their colleagues, friends and families that their old phone isn't just faster than new ones but also safer and cheaper, shows no ads, doesn't steal personal data, doesn't require constant updates (==cheaper data plans) and supports the same software they could have installed on their PCs. Definitely, phone (and their OS) makers would hate that.