I want to take tripod pictures of small remote objects such as birds with a 300mm prime lens. Would the picture be degraded less by adding a 2x telextender or by cropping and magnifying the middle quarter of the picture?
Do you have the TC available? If so, why not perform an experiment & try both methods? From the examples I've seen & comments I've read, a 300mm prime (at least Canon's) is supposed to be able to handle a 2x TC pretty well.
I would not suggest using TCs with zoom lenses as the results are less than optimal.