Report mentions that the a/c was frequently, or constantly, out of trim.
737-800 is sensitive to control wheel pressure/inputs while trying to engage the autopilot. I'm not sure if an out of trim, or significantly out of trim, a/c will allow the A/P to be engaged. Maintaining attitude would require force on the yoke and releasing it would display it from neutral as it seeks the trimmed speed.
Automation 101 - push a button and VERIFY that the correct action took place. No assuming, verify.