Since it's not staying running even with starting fluid, I would check compression. It's quick and easy.
Then change the boot. It's cheap and the tiniest of cracks will ruin it. Scrub the carb and manifold with a wire brush so the sealing surface is nice and smooth for a good seal.
Past that it sounds like a fuel issue. Remove the carburetor, remove the fuel bowl (carefully), remove the cap (carefully), remove the slide assembly (keep needle/plastic/spring in order). Then spray the living hell out of everything that's not rubber with a can or two of carb cleaner.
Put the slide assembly back into place and the cap on top (careful not to pinch the diaphragm). Replace the fuel bowl. Snug the screws down securely, do not over-tighten them.
The screws on the fuel bowl and cap are soft japanese screws (Keihin carb). If you don't have the proper JIS screwdriver, being very careful with a #2 phillips. Or you can use a dremel/hacksaw to slot the screws and use a flathead driver.