It looks like all the contacts are well made. The prongs both get a significant stretch out when the plug goes in. I'll give them a little rough up and push them in some more to eliminate bad contact.
Shorting tip to barrel on a guitar lead generally is to be avoided like the plague. That's the godawful pop you get pulling a lead out of an active channel, no way I'm doing that deliberately.
The sound is either silent or a minor glitch when it goes out.
Is it possible for the barrel to lose contact without corrosion? I haven't considered that at all.
It sounds like the contacts have lost some of their springyness. This is a common fault. The usual quick fix is just as you say to bend them in a bit more. But usually that won't last if they've already lost their spring.
Other possible faults are that since these jacks are sort of crimped together sometimes the crimp can get lose and the parts short when you wiggle the plug.
Another is that the prong pressing against the plug tip is possibly shorting against the conductive shielding as the insertion of the plug pushes it back away from it's normal position.
And of course there could be a bit of corrosion.
However new jacks are so cheap that usually unless it's an emergency situation best is to buy a nice new switchcraft jack (they are much less prone to losing spring) and install it.