Sounds like its idling lean (both from the pumping and especially stumbling upon acceleration.)
As a diagnostic test, try idling and revving it with the choke manually applied. Have an assistant block off 75% of inlet, either by choke valve or even fingers! If it smooths out, you have your culprit.
Also pull the plugs and read em.
If it turns out it idles lean, you need to find out why. Blocked jet, leaking gasket are possibilities. Also the Byzantine feedback mechanisms.
P.S. Is your Carb a Constant-velocity type? If so, a leaking CV diaphragm is quite possible.
Good luck!
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