Handing over your card to the waiters is standard in the US. You are basically trusting the waiter to not skim/copy the card. In the case that it is skimmed, the fraud policies I've experienced are very favorable to the consumer.
I was surprised to see in Quebec City that all the stores/restaurants had portable card machines they brought to you. That does make a lot more sense to me.
It's "the way that they do it" in most places at least where I have been, that is giving the waiter the card and wait for them to come back.
In Sweden restaurants have started to come out with portable devices and you even have to put in the amount you want to pay (because they want tip which is already included in the price..). I find this really good but at the same time less convenient because I don't want to fiddle with their device.
I would feel much better if I had my cards stored "securely" in my phone and could do "Tap to Pay" over NFC instead and have the waiter bring over an NFC device to tap against my phone.
Paying by card here happens either by using a mobile card reader, or by just getting up and walking to the cash register before leaving. Much safer.