Sometimes it isn't that the pump is bad, it's that the small passages inside the carb get cloggged. If you're careful (and lucky) you can pull the carb apart without ripping gaskets. You do have to inspect everything and make sure the rubber and paper parts aren't dried out/brittle/cracked, but possibly the carb just needs to be cleaned inside.

There are a lot of small passageways inside the carb that, even if you do rebuild it will need to be thoroughly cleaned out. If you did put a rebuild kit in it without cleaning it all out it would be entirely possible that the problem doesn't get fixed.