What your saying now is not what you posted. I don't care where your scenario takes place, all things equal if you have grappler versus non-grappler the outcome generally goes in favor of the broader game.
I came across a book detailing the numerous boxer vs grappler confrontations throughout history. Boxers generally came out worse but not by as big a margin that I thought. It was about 2-1 in the grappler's favour.
The person doing the stomping would be a stranger in the crowd. you are someone with no grappling experience at all probably
Rightly said by you that Boxing is a proper sport not just a fight. Its not like MMA that someone stands & fight for a ten to fifteen minute max fight which can be done by some hard wrestlers also. But Boxing is a skill & such an art which needs years of training.
You know what MMA stands for? Do you know what boxing is? It's a latter 2/3rds of MMA Hockey is not Ice Soccer.
Ice Soccer! HA! Laughed out loud at that one. No sarcasm. I'm a Judoka but still would take boxing. We have to remember the preeminence; yes boxers finish more street fights but they probably get into more street fights haha, different crowds do the different sports, and the temperament of judo players is to have much higher patience and self control than the average boxer. Just my take.
I don't see boxing as a martial art. It was never designed to be a martial art, boxing is competition and a way to prove who is better in a certain ruleset. If it was a martial art it wouldn't limit itself with rules. The thing is that boxing proved to be one of the most effective things in real confrontations, but that's precisely because of its character as a sport that relies on sparring and strenght & conditioning. By this logic you might as well consider sports that rely on explosivness and strenght like rugby a martial art because being a well-trained rugby player would no doubt give you an advantage in a street fight against most people.
The sport - it's origins are all about self defence. If a boxer put a pair of knuckle dusters on or his car keys in his fist and goes to work is it a martial art then? Every martial art whether it be eastern, western, northern or southern has rules. Some have athleticsm and heavy contact competition as their core thus define their rules more expansively and compete more often i.e. boxing, MMA, muay thai, bando. Interesting you mention rugby as I consider them one of the most dangerous opponent's in the street. We have a few Pacific Islander ancestry Muay Thai fighters who also play Rugby at local level - generally very tough, athletic, good balance and calm under pressure.