Rain generally also comes with a lack of sunlight. Cloud cover for an extended is just as bad as shade.
Let’s say your hypothetical golf course is in the Pacific Northwest. How much golf traffic and inhibited photosynthesis do you believe the turf will tolerate in an unusual winter?
Let’s also suggest your hypothetical golf course has a hundred different micro-climates which most people also find ideal for golf.
Poa has several competitive advantages over other turf varieties because, as Tom noted, it is an annual plant. Poa annua seeds are viable in the soil for years and when all of the above conspire to compromise a healthy stand of anything else the Poa is already there to germinate, live out its annual life cycle, throw out more seed, and die.
That being said the answer is almost always topdress, and then topdress, followed by some topdressing.
Not for nothing, but where the climate is appropriate I believe that Poa annua are the finest putting surfaces in the world.