As of Brave New World, Culture works differently: beyond the Monument, each Culture building provides a trivial +1 Culture, plus slots to hold Great Works. Don't think of an Amphitheater as providing "+1 Culture." Think of it as providing "+3 culture and +2 Tourism once it's full."
The point of Amphitheaters in Civilization 5 is to hold Great Works of Writing. There is no other building that can be built in every city to hold them. The other buildings that can hold Great Works of Writing are the National Epic, Heroic Epic, and Oxford University, for a total of four slots. If you're quick, you can also put some Writings in the Great Library and Globe Theatre, but in my experience the Great Library is always built – very quickly – by one of my opponents. (I usually play on King or Emperor, and the head start that the AI gets is enough to get one of them to Writing several turns ahead of me.)
As with other Culture buildings, it's worthwhile if you want to achieve a Culture victory, but also if you want to unlock social policies faster. If you use a Great Writer to write a political treatise instead of a Great Work of Writing, Great Writers generate about 8 turns' worth of your current Culture output. If you don't care for a Culture victory, then you can measure the Culture generated by a Great Work until the end of the game versus 8 times your current Culture output to see which one will get you more Culture.
Of course, building Opera Houses and Museums requires you to have an Amphitheater first. You won't always fill up your Amphitheaters, but once you get Archaeologists, it's very easy to fill up Museums with artifacts. Skipping Amphitheaters implies that you don't care about culture at all, which means you will have fewer Social Policies available to you. (And if you have only one high-Culture enemy, that enemy will win a Cultural Victory relatively quickly!) I find Social Policies to be so useful that I wouldn't consider ignoring them.