If monuments are obsolete surely theatres are available!!??

Or at least drama will just be a 1 turn tech by then.
Music is perhaps more likely to have been skipped and is a more expensive tech, but if you have it, building culture is the cheaest way to get a border pop just to claim a resource - 10 hammers. That's not ideal though for culture wars since you need something to keep "pumping" culture.
By astronomy I find I'm often in caste system, and an artist is quite efficient then. Say you take a citizen off a notional average 4 hammer tile - it's cost 16 hammers for the border pop. It's also a more convenient way of generating ongoing culture for a "mild" border war since the city can be building something else, and doesn't have to dedicate its queue to building
I'm not sure about the efficiency of religion if it's just for the culture. It's 40 hammers for the missionary, or cross fingers for random spread which isn't very convenient. If organized religion is active though it's great because not only do you save on the monument you're magnifying the speed the new city can build its infrastructure. 1

is pretty puny for an ongoing culture war but it's something, and enables the religious buildings, which the AI will certainly be building in its city!
I remember reading a long time ago that the monastery was quite an efficient source of culture - looking at the costs now it's OK but not brilliant. 60 hammers for 2

which is 30

/

, the same as a monument. But better than library and much better than a temple though. Theatre is 50 for 3, or 17

/

which is both the strongest and the the best value culture building excluding the later +50% ones.