It is very possible to soften or lighten a horse's mouth and make him more responsive, particulary one that's been taught to "tune out" subtle aids, like an old schoolie has.
In this case, all his previous inadvertent "training" has been to ignore all but the most obvious, unambigous aids, and only pay attention to big, crude, rein aids, preferably those that are supported by weight/voice/instructor's voice.
Reschooling is simple, as long as you are absolutely consistent. Ask once, with the tiniest, subtlest, flutter of a butterfly's wing aid. Ask for the second time with a soft to normal aid. Ask for the third time with an OH MY GOD, the world is going to end, we're all going to die, mouth open, whatever it takes, DO IT NOW aid. Repeat. Often. Lots of upward and downward transitions in lots of situations. No excuses. Even if Martians are landing in the ring, if you don't get immediate obediance by the second ask, go for the end of the world as we know it aid. If you are consistent, your horse will start being very, very attentive, listening for the flutter of a butterfly wing aid.
It can be helpful to change to a bit that has a slightly different action - Dr. Bristol from a plain snaffle, three corner from a smooth, double mouthpiece from a single - not necessarily harsher, but different - to help him tune into the new expected response.
The only other caveat is that is if you let him be ridden by beginners, or even a careless experienced rider, he'll revert to tuning out, so be vigilant.