You really could have, you know. I brought one every session of every day (among other forbidden items like Tim Tams) and no-one even noticed.
I actually did have an empty bottle sort of hidden and they made me throw it out. I realized if I hid it deep enough they probably wouldn't notice it but by then I didn't feel like trying to sneak it in, but my mouth got very dry throughout the week.
Well, it's not normal in my experience, but to my disbelief I found that to be the case everywhere in Boston, except for the airport.
Interesting, I have a tendency to avoid public restrooms at all costs so I didn't really notice the lack of seat covers throughout Boston, but I thought they were standard for large arenas/venues since the bathrooms are used by so many people.
I know that restrooms in Asia don't usually have seat covers, but I found that many of them will have disinfectant that you can at least use to wipe the seats down before use (or in the case of Japan, they'll have a bidet or squatting toilet).