It can't be the current image of the pool player that results in no sponsors of pool. Who wouldn't back a field in baggy shorts, hat on backwards and one arm tattooed like a comic book page?
That describes most popular sports (NBA, NFL, MMA, etc) that do great with sponsors and have million dollar paydays for top athletes.
I think the difference is that pool is a bar game in the US, which isn't something kids can be encouraged to do (no one is going to drop their kid off at a bar after school). In Europe, pool doesn't have the same ties to bars/drinking and is often played in clubs that are more suitable for children.