Welcome to JUST ANSWER small engines, let me try to help you !!
Fuel pump or carburetor is not getting enough fuel to let engine run under load. It takes more fuel to run under load than without load. Start with removing the carburetor float bowl and check for debris in the bowl and check the float and needle to see if the needle valve passage is open and not full of trash. Here is a basic carburetor link.
Try this and we can go from here. Thank you.
Ok, let me know, we can continue from here. Thanks.
Yes, the pulley is a variable pulley and will close down on the belt as soon as the engine rpm comes up. unless the belt has pieces missing, it should not cause jumping.
The pulleys change shape as engine and speed changes.
Ok, so the engine runs! That was what I was trying to find out. Then you drive clutch is bad! The clutch and belt should be replaced.
The pulley is on the tapered end of the crankshaft and takes special puller to remove it.
I would go with a new drive clutch, not try to rebuild.