In defense of how you were originally taught, it is really hard to count out loud while playing trumpet. If can begin doing counting out loud while playng guitar, it will make it infinitely easier to learn new music. Otherwise, it is really difficult to figure out why something didn't sound right, especially when dealing with 1/16 notes where you can easily lose track.
Hard to respond without stating the obvious - that a mouthpiece pressed to the lips requires other methods. Such as getting the beat internalized before starting to play. Sort of like visualizing hitting a drum pad on the beat.
Most of my inaccuracies are not caused by missing the beat. It’s from missing the right note, hitting too many strings, or late on quickening chord changes. I did count out loud on one part of the strumming exercises but no longer feel the need to do so.
To quote John Nemeth: if it ain’t broke, don’t fix it.
BTW: “Too many Dirty Dishes” is one of my favorites. Pots and pans everywhere.