But if they didn't try to push the nose down, they wouldn't know how much or how little authority they had.
From the looks of things, the pilot only put the nose down once, and the airplane did accelerate over 60kts, which then re-activated the stall warning.
From then on out it looks like the stick was held back, nose up.
It doesn't seem to me that a lack of an attempt to use the manual pitch trim was a factor, as they didn't try a nose-down stall recovery that may or may not have worked due to the trim setting.