In any game that isn't early Domination, I think the Hermitage is worth building in your guild city to speed up social policy acquisition. For that, you need amphitheatres and opera houses. Otherwise it would certainly never be worth building opera houses, or even more than a couple amphitheatres.
Hermitage is a tough cookie. It's great but it has a such an enormous cost that you want to have a really good reason to build it.
On Amphitheatres - I agree with the consensus here. They are probably one of the worst buildings in the game and all the more annoying because you need them and the equally useless Opera House if you want a Hermitage or a spot to store writing/music. But something else that people here haven't mentioned is that working the musicians guild is a great way to build extra culture (even if you're not going for a CV) and once you're able to build Opera Houses & put Works in them it really helps with late game culture.
Hence that's why I value a free Amphitheatre from legalism even more. It costs an extra 60 hammers (so more than double a monument cost) for half the yield thus it is absolutely terrible to have to build - so why not take it if you can get it free?
And I always make at least 1 Great Work of Writings before bulbing them for culture so with a Great Work of Writing I at least get some value out of the building.
If you bulb 2 can swap them into the Oxford University to get a total of 6 culture & 6 tourism so the value goes up a bit higher. This gets better if you do decide to build Hermitage, Sistine Chapel, BroadCast Towers, Sydney Opera House.....
But if I'm going to whinge about Amphitheatres it's that they don't at all come close to doing what they did in Classical Greece. They were the Greek centre of culture and entertainment - the equivalent of 'the movies' in today's world. But in BNW it's just an annoying building that you
have to build if you want to get to a Hermitage. It's role really overlaps with the Colosseum and in the ancient world there really isn't a practical difference between a gladiatorial arena & a drama theatre. The building is just a pit or stage surrounded by seating for spectators