David, you are not wrong about being able to improve on the fuel consumption and ease of starting of the Briggs 80202. Because it is a first generation Briggs design your carburetor has a really nasty choke, and like all side valve mower engines I can think of, it has a compression ratio of only 6:1 so fuel consumption is relatively high. However despite the nasty choke it should always cold start with three pulls. Pull once or twice with the choke closed and it will chug then rich-load and stall. Then open the choke wide and it will start first pull and run until the tank's empty if you don't turn it off. Also, despite the poor thermal efficiency of low compression engines, because a reel mower has a low power requirement compared with a rotary mower, you'll still find the cost of fuel is extremely low.
There is one engine that will outperform the 80202 on both starting and fuel consumption: a small overhead valve Honda from the days when they were still made in Japan. If kept in perfect tune it should start first pull, and with its compression ratio of 8.7:1 for early production and 8.0:1 for later production, it will probably use 20% less fuel for the same power output. However it would look wrong on your mower, and pretty much eliminate its collectibility. Parts are also notoriously expensive, and professional repairs prohibitively so. Since you would be buying the engine second-hand, it sounds a risky proposition for trivial gains over the engine you have.
Of course if you want a more advanced engine, you could simply fit an OHV Honda and put the Briggs into storage so the mower can be restored to orginal condition at some later time. Because the engines are physically interchangeable, this is easily done. However I suggest that you avoid Chinese-made Hondas, and even more rigorously avoid Chondas (unlicensed near-copies of Hondas made by Chinese firms). Also avoid side valve genuine Japanese Hondas, because they have no fuel consumption advantage over the Briggs.
