That's certainly not the norm in the US. I've eaten in restaurants in every state and have never seen or experienced anything like that. It sounds like you just got a terrible server or restaurant.
Obviously you will do as you please, but I wish tourists would try to get with the culture. As a European, one of many reasons I don't tip is that I don't want to encourage a class system where anyone who doesn't pay up is seen (and sees themselves) as stingy.
Also about the $10 - they would need to gather quite a few more dollar notes and spend time finding a cash exchange bureau. Else the note is just a memento of a weird American.
In some EU countries the tips have to be counted as salary and income (+social) tax paid, or even they have to be declared as revenue to the restaurant along with the VAT (19-25%) going straight to the state. Of course, when paid in cash - that rarely happens. The point is that the provision exists in the law (so the tips are not unheard of)