Because it became quieter when you fitted a different connecting rod and bearing shells, it appears at least a substantial part of the original noise came from the slightly damaged big end bearing. The remaining noise could be because the replacement bearing also had some wear, or it could be piston slap. When an engine is cold, I personally have some difficulty distinguishing between moderate big end noise, and piston slap. Piston slap, however, quietens when the engine warms up and the aluminium piston expands. If the noise remains the same on your engine, the remaining likely causes are:
1. Big end noise, due to a combination of a different size-grade of connecting rod, and some degree of wear to the crankpin and big end shells.
2. Tappet clearance. I tend to expect the older side valve Briggs engines to have noisy to very noisy tappets. I do not have experience with side valve Honda engines.
3. Main bearing rumble.
These three types of noise have different characters, and can be distinguished easily if you have prior experience of each.
I agree that it is unrealistic to expect an old side valve engine, rebuilt with worn parts, to be mechanically quiet. If the noise sounds like other moderately worn side valve engines, it is to be expected and should not prejudice the continued satisfactory service of the engine.
Please keep us posted on the rebuild of the smoky engine. It might be worth attempting closer clearances with this one, to make it quieter as a sort of training exercise. That would imply attending to big end, piston skirt and main bearing clearances.
