Running programs inside a sandbox is not what the OP is talking about. If you can't reinstall your own OS and get root permission* to talk to the hardware, you're not really in control of whats going on.
*Im of course aware you can root Android phones and jailbreak iPhones, again this isnt what OP is talking about - they void warranties and are challenging/time consuming even for tech people.
Back when I first installed Linux on a PC it was much more challenging than putting LineageOS on a phone. Especially getting full hardware support on a laptop took a lot of fiddling.
Nowadays, Lineage will work perfectly OOTB 99% of the time. And 100% of the time if you buy a phone which is already compatible.
Why do you need root access to the hardware for it to be considered a general purpose computing device? If someone provides you an account on a server where you don't have root access, would you not describe that a general purpose computer device with a real filesystem where you can run your own programs?
The manufacturer gets to decide if they're manufacturing a general purpose computing device or not, but they don't get to define what general purpose is. Either it executes arbitrary code or it doesn't, which my Android device does just fine without root access. So I would call it a general purpose computing device.
Root access is just a platform specific construct, and it doesn't necessarily have an important purpose on every platform. Furthermore there are lots of ways that a platform could take away your control even if you had root access, such as through binary firmware blobs (which I'm sure you're running dozens right now, as am I probably).
But there are lots of situations where you can't do things even with root access on a unix-like machine. I gave one example already -- root access doesn't give you control over the operation of system firmware. But there are others, too. For example, root access doesn't let you write to kernel memory, it doesn't let you bypass SELinux policies, etc. The way you are defining root access here essentially doesn't apply to any modern computing system.
Isn't your software the thing that's really "going on"? Who cares about the intricacies of how the hardware works? I understand being interested for hobbyist reasons, but is there a practical reason?
EDIT: Here's another example. Let's say you have a VPS on which you have root, but you don't have ring 0 access on the machine where it's running. Are you in control then?
if the remote server has better IO and net access, but if you chose to use your own software, the vendor decides to cap you access (or force an inferior api on to you), then you have an equivalent analogy.
So if a general purpose computing device comes with a camera, but the camera is restricted, then it's not a general purpose computing device anymore? What if it didn't come with a camera in the first place? Plus, there is no restriction against connecting your own camera peripheral which you do control.
The first time I rooted a phone and installed a custom ROM completely over the air while riding a bus I felt like a hacker god, but that was like 2012 or something. I'm honestly surprised everyone doesn't root their phone because at least on Android it really is trivial these days and fixes so many problems with the phone ecosystem. There's no excuse at all for someone honestly interested in running their own programs not bothering to figure this out.
*Im of course aware you can root Android phones and jailbreak iPhones, again this isnt what OP is talking about - they void warranties and are challenging/time consuming even for tech people.