Months later, I was doing some research on basketball player Frank Kudelka, and player from the Bay Area. I happened to notice in a box score of one of Kudelka's basketball games while in the military that there was a player on a Coast Guard team named "Gallette." Noticing that "Gallette" is similar to the unknown Washington Capitols player "Gillette" and also from the Bay Area, I looked into this possibility of being Gillette. Gallette played for various teams around the Bay Area, but was he actually Gene Gillette?
And then I found this obituary from the San Mateo Times:
Looks like Gene Gallatte is the name of the mystery player, not Gene Gillette.
I am going to take a second to pat myself on the back.