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.
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).