And ten minutes should certainly be enough to do that. For that matter, if you're constantly pulling back the yoke to stay level... Wouldn't you try setting the trim manually?
Investigation of the wreckage should clearly show what position the stab trim was at. It's a large nut on a jackscrew, the disintegration of the airplane should leave that intact. The nut/jackscrew is built out of incredibly strong steel.
Pilots have long been taught that pulling back on a 737's control column can arrest that condition – a fix pilots call a "breakaway", says the APA.
Indeed, American's runaway stabiliser checklist, dated 10 July, says, "stabiliser trim commands are interrupted when the control column is displaced in the opposite direction".
But the APA has now learned that the 737 Max is apparently different.
https://www.flightglobal.com/news/articles/airline-industry-...
As has been stated above, stab trim runaway is common knowledge and an emergency procedure (EP) that is trained for. Arresting the descent with yoke while activating the electrical stabilizer trim cutout and having the pilot-not-flying manually rotate the stabilizer wheel in the opposite direction or at least attempting to hold it from adding more trim if cutout is not working.
Many pilots will position a knee or shin on the stabilizer wheel so they are aware when trim is being run. The yoke itself holds a trim adjustment mechanism so the pilots can adjust trim while retaining arrest authority on the yoke as well, though in the case of runaway trim the cutout would be used and manual operation of the wheel would be required.
I believe another commentator mentioned the jack screw, this will very likely have survived the crash and investigators will be able to tell exactly where the nut was positioned (trim setting) when impact occurred.
With stab trim runaway being such well trained for EP [assumed as its common in my circles] I would be surprised if that were the root cause unless pilot error is attributed to suspecting another fault mode.
I am not familiar with commercial aviation as much, is it common place to transition from the critical phase of flight during take off to autopilot (AP) climb at such a low altitude? If they were not on some sort of AP mode I can't see how they wouldn't notice the trim being added as a serious issue.
-Sorry I had meant to post this as a response above and not directly to your post
There's a fair number of people who think that pilots are relying on "George/Otto" too much and should do more hand flying, but I'd imagine if you peered into 100 airliners on departure, that 95 or more of them are on autopilot somewhere below 5000' AGL.