How gun size affected age-of-sail rate of fire

Did smaller naval guns fire much faster than larger guns in the age of sail?

Only somewhat. Manual loading and running out the gun limited the advantage until built-in breech-loading weapons of the ironclad era made caliber a much larger rate-of-fire factor.

A lighter cannon was easier for its crew to move through the loading cycle. Even so, every muzzle-loading gun required many people and much of the same sequence of labor. That kept the age-of-sail rate advantage smaller than a simple weight comparison suggests. Breech loading and fixed mountings in the steam-and-iron era changed the mechanism enough for gun size to produce a more pronounced difference.