Does eating meat require rinsing the mouth?

No, it is better, it is better that a person, in the beginning, during the time of the Messenger ﷺ, at first, wudu was required after eating anything that had been touched by fire, especially meat.

Anything that had been touched by fire, they would make wudu after it, then later on, the last ruling was to leave wudu.

From what had been touched by fire, the scholars say, so the obligation was dropped and only recommendation remained.

Meaning, anyone who eats something touched by fire, bread,

rice, meat, anything cooked.

If something has been cooked on the fire, it is better for a person to make wudu, but it is not obligatory, it is just better.