Thanks for all your suggestions. With your help I was able to figure
out the problem. The cap was bad! I tried nudging with a screw driver
after disengaging the gear and the motor started spinning. The root
cause was different though. The bottom part of the motor housing was
cracked (one screw that was holding the bottom housing was loose, that
is how I noticed it). Because of that the gear on the motor shaft was
resting on the bottom housing instead of the bearings (in fact this
might have killed the cap). I tried balancing the motor with a metallic
string, it might have worked even with the cracked housing. In the
process of trying this out, I caused the limit switch to go beyond its
max. So I dropped the plan to fix it. Even otherwise rather patching up
the motor housing and finding another starter cap and putting
everything back together (which might not last long any ways) I am
planning to get a new one.

Any suggestions about new GDO? A couple of colleagues suggested
Chamberline, belt type. I might get that one from menards. Still
debating whether I should install it myself or get it installed
professionally. I do not have more than a couple hours to work on this,
so if it will take ~4hrs as some people said, I might as well pay for
getting it done.

Thanks again for all help.