I've read of a leaking carb float causing too rich rough running. Carb heat warmed the air in the float, causing expansion, which pushed fuel out, restoring float buoyancy, and engine ran O.K. This lead to carb ice being assumed to be the problem. Cold air lead to contraction of the vapour in the float, and more fuel entered the float, until the mixture was so rich the engine lost too much power for the aircraft to remain in flight. The pilot made an off-airfield landing, blaming carb ice.
Would leaning the mixture work with a sinking carb float?
Sounds good on paper. However, the carburetor heat doesn't come near the float, and doesn't warm it; carb heat is used to adust the tempeature of the induction air...not the carburetor, not the float bowl, and not the float. The float is sealed, not full of fuel, and the temperature of the float doesn't make any difference in it's ability to function. Regardless of it's temperature or that of the fuel, it should always float. The function of the carburetor float is only to stop fuel flow into the float bowl, when it reaches a predetermined level, and it does this by action of a needle valve actuated by the float. The float bowl, in turn, only serves as a holding resorvoir, a small "header tank" if you will, which supplies the main and idle jets.
Worse case senario, you have a float which fails and leaks...preventing it from floating on the fuel and shutting off fuel flow to the carb float bowl. The carburetor floods out.
A certain amount of air above the fuel in the float bowl is required, as the main jets work based on differential pressure between the carb throat venturi, and ambient air pressure/fuel pressure...the ambient pressure supplied by venting the float bowl. Flood it and you can have an engine stoppage, or supply enough fuel uderpressure adn you can flood out the carb and engine...and have either a fire or an engine failure.
Can you prevent it by leaning the carburetor? Possibly, but don't count on it. You can go to cutoff on the mixture and still have a fire hazard or flood problem. Where it's usually noticed is when you approach the airplane to preflight...the float has sunk and you'll see fuel leakage down the nosewheel or through the bottom of the cowl, with stains or wet goo present depending on the nature of the leak, the ambient temperature, etc. In flight, you're not going to know the float has sunk; you may have a rough engine, you may not.