I think you did smart, having physical record just makes it easier to attempt fraud versus the swiping that goes on for almost every transaction these days.

In generally in the US most eating establishments users think nothing of giving the CC to the vendors and they take it away 10's of minutes, who knows what gets lifted, including the 3-4 digit security if they have dishonest employees. From a risk point of view a bigger deal with more information leak.

CC fraud still is rampant but most major CC also have aggressive AI tuned fraud monitors so actual loss to you is slim to non-existent but the hassle of fraud reporting and getting a new card are a PIA if traveling.