The "spread culture" mission is, in my experience, useless. Yoy can't perform it unless thereis already some of your culture in the city, which requires one of your cities to be culturally powerful and close enough to the target for the latter to be within your city's cultural radius, and the incremental effect of spread-culture missions is negligibly small. I once tried sending hordes of spies to see what happened, amd after several dozen missions (at ever-increasing costs in EPs) my culture was still rising by only 1 or 2% per turn - which trend continued after I got tired of sending spies, thanks to the culture from my nearby cities.
Anyway, even if the enemy city is totally surrounded by your territory, is weakly defended, and does not have its owner's state religion present, it still may not flip. There is only a 10% chance per turn that flippability will be tested, and there is always some chance that even then flipping may not occur.