If you think that's gross, I've seen baby roaches in the parmesan cheese shaker that's on a pizza hut table before. Any of the condiments out on a table at a restaurant are most likely a wee bit questionable. Once I left my last food service job, I made sure to tip better, and rarely went to restaurants for a long time.

with a napkin

doesn’t kill bacteria but kills cooties

If you're that concerned, I'd bring them. Do you wipe down the bottle? Because people definitely don't wash their hands and then they touch it. Or lick their fingers and touch it. Even if the spout is untouched, the bottle is well-handled.