True, if Arnold had died at Saratoga, they would have named cities, schools, universities, streets, parks etc. etc. etc. after him. "Benedict" would be an extremely popular boy's name. There would be a huge statue of him in Washington, DC.
But he made a very bad choice.
Not just Saratoga. He fought courageously in Canada, as I recall, and one of the reasons for his betrayal was that he thought he was not getting the recognition for his efforts that he was entitled to.
I’ve often thought that the reason Saratoga/Ticonderoga has not been much more greatly celebrated as the turning point in our revolution is a lack of a suitable hero as a focus of attention - Gates was an embarrassment, Arnold later became a traitor, and Washington and other key leaders were not there. So it’s difficult to celerate such a triumph on a national level without anyone to ‘personnify’ the triumph. Why else would such a crucial battle, which evidently swayed France to support our cause, be almost unknown to the general public? It’s certainly very well-known to historians, but when I have mentioned it to friends or acquaintances I have often drawn a blank stare - What battle? Where?