Hi and welcome to mightyram. Sounds like the accelerator enrichment pump on the carb has failed. Easy to check (engine off) - remove the air filter, manually hold open the choke butterfly and twist the throttle linkage open and closed a few times. While you do this, look down the primary throat for a jet/stream of fuel coming from the side of the throat. No fuel stream or a dribble of fuel indicates the pump diaphragm has failed. You don't need to remove the carb to fix this (a little fiddly to work on while the carb is still mounted but do-able) but as the pump diaphragm is part of a kit I would say it's time to overhaul the carb anyway seeming it has been sitting for so long. They probably installed the electric pump thinking fuel supply was part of the problem but be aware that if the engine starves out of fuel while at highway speeds or being under hard open throttle it may indicate the mechanical pump is struggling.

As for your question - yes the Weber swap is worth the effort. Not a straight forward deal but the minor mods that need to be done aren't complicated to pull off. The Weber will be happiest with accompanying engine mods but will work fine just as a change over. Depending on the electric pump you are already part way there for the conversion (the Weber needs a high volume, low pressure pump to operate correctly) You have 2 Weber carb options - 32/36 DFEV or the more modern designed 38DGES. The 38 has more scope for engine performance tuning. Both will offer better power delivery and economy than the stock carb under daily driving conditions.

*engine type - if your engine has an alloy timing cover it's a 4G54/G54B 2.6 otherwise it'll be a belt driven G63B 2.0 engine.