I take the view of buildings as concepts which evolve. That temple you built to Baal or whoever in the BC's in the real, modern world would have been demolished long ago and replaced by a new building dedicated to a new god, often erected on the foundations of its predecessor. But there will always be a venue for worship, providing that 2 culture.
I look at the circus the same way. Those that survive indeed serve a cultural purpose now, but the circus doesn't grant any... You could apply the same logic to any ancient building, which will never retain its original function, if it survives at all. So for the amphitheatre, it just becomes a stadium in my mind and the modern one is just the supersize version. You have so many of these for different sports anyway.
As happiness is in shorter supply, I would change the circus to 1
1
to allow for other buildings (such as the public baths) to grant happiness without upsetting balance. Gold: I'm sure betting is of all times. Besides there are already the metro and airport in the late game that produce gold and even without them, your income can get extremely high.
I see your point about the horses. But they DO got obsolete for most of their (non-cultural) functions in real life too. Racehorses have a pedigree and their numbers can hardly be related to the masses used in the military, transport, farming, ... But I agree seeing e.g. 20 horses appear in the resource list feels wrong (smaller numbers are ok for me), thus my alternative. In the ideal case horses should go obsolete and disappear from the resource list, just serve as a bonus resource from that point.