You seem desperate to make out like feet on seats is so much worse than most other small things, but we all know that this requires some serious stretching.
A bit of loud pop music is irritating.
I believe that putting feet on seats is much more destructive, much closer to etching or graffiti than making a bit of a racket. YMMV.
Children respect people who give them respect - this is the most important rule that anyone in charge of them has to learn, and they have to learn it quickly. Physical assault by a much larger individual doesn't teach respect: it breeds resentment.
In more prolonged interactions I agree. Teachers who command respect have more control than teachers who try to rule with an iron fist. It's the same with any leadership position, both with children and adults.
However this situation is different. The old guy asked them, nicely and repeatedly, to take their feet off the seats. They took the **** out of him: as the eyewitnesses say, they were "being cocky" and refusing.
I don't see where respect comes into it. Asking nicely didn't work. Walking away would be better for the blood pressure, but would also teach them that they can get away with things if they're a smartarse. The way the man reacted wasn't the way I would have, but I'll tell you something: the kid who was bawling his little eyes out took his feet off the seats, didn't he? And I bet he'll think twice before he's a cocky little scrote to the next old man.
ETA: It's not the way I would have reacted, and I can't condone it: life's too short, more than anything else. But all this talk of "violent assault" and "locking him up" for "child cruelty" is absolutely ridiculous. The old guy didn't hit them and didn't leave a mark. Maybe the police should have a quiet word with him, right before they send the kid's parent an £80 Penalty Fake for breaking the railway byelaws.
There are prison wardens in Australia accused of stripping children in their care naked and teargassing them: now
that is child cruelty.