The neck and back pain is a legit problem for some people. You do see surgeons, particulary older ones with stooped over shoulder a lot. I know a surgeon that even retired early from it because his back was bothering him so much and he just couldn't take it. That being said, it isn't always because of height. As noted, the tables are adjustable. Some of it takes good habits, stretching and all sorts of other things. Also, many people regardless of profession suffer through back and neck pain. When you enjoy what you are doing, you tends to be easier to ignore the pain.
You aren't too tall, but you'll just have to gauge whether you can go through all those years of being stuck with a lower table. Also, some people may use loupes to see better when the table is lower. I'm not a surgeon, so I can't actually testify if that is a common technique.