Being a deeply entrenched member of the "Tecumseh Tribe" and a fan of flatheads in general, I offer the following discussion of your problems with the Tecumseh .
The Briggs install took 3 minutes, how long did the Tecumseh install take and what made the difference?
It took 50 pulls to start, how long had the engine been sitting? The diaphragm carb is admittedly sensitive to clogging from old gas. Briggs are less so. No engine will run if old gas is left in it. My rule of thumb, if an engine does not start after 5 to 10 pulls with fresh gas (assuming it has spark and compression), then it probably has a fuel delivery problem. At that point, save your shoulder, stop pulling and starting looking.
The engine can easily be configured for a minibike throttle cable with the part no 730136a throttle control. Its sounds like the Briggs was already configured for minibike use?
The exhaust can be redirect by replacing the stock muffler with a 1/2" 90 degree street elbow and a hot dog muffler
The leak in the carburetor is probably due to a bad diaphragm, which was probably the reason for hard starting. Diaphragm carbs are kinda tough to adjust properly. Float carbs seem easier.
As time passes, I'm leaning more and more towards, the cheap Tecumseh replacement carbs. Its becoming more difficult to find good original adjustable rebuildable cores. Good rebuildable means those which rattle ie., have UNSTUCK idle fuel passage reduction rods. Some times they can be freed up, but most times not. In MY experience, the engine will not idle right with a stuck idle rod. They will run like crazy with the throttle open but wont idle. Serviceable throttle and choke shaft openings as necessary as well.
Another reason is the time factor, I have a lot of engines to build and with every passing day, less time to do it. A $12 plug and play carb is a big time (and shoulder) saver. I prefer original carbs whenever possible but sometimes its better to swap it out and "keep it moving" .
Its interesting that the Tecumseh outperformed the Briggs since the fuel set up on the Tecumseh has an intake manifold passage the size of a sippee straw.
A final observation, a 3 hp engine seems awful small on that size of a bike, looks like a Tote Gote of some type. Kinda like a trolling motor on the Titanic.
A lot of these ideas here are rehashed , but still looking to forward to any technical discussion of these points.
Thanks for keeping flatheads in the fold.