Interstate Love Song was released as a single in August 1994, Scott wasn't busted until 1995. Scott's addiction probably had nothing to do with it.
STP made two late night appearances in 1994. In June they appeared on Letterman and performed the first single off the album, Vasoline. In December they appeared on Letterman again, this time performing the fourth single off the album, Unglued.
So really, the reason ISL was never played on late-night was because they just weren't booked for a late-night show at the time Interstate was released. Had they appeared in August instead of June, they would have played ILS instead of Vasoline... However, because of marketing they would have wanted to play on late-night as close to release as possible, so performing latter to promote the second single instead of the first would have been a mistake.
All that said, they did play Interstate Love Song on Jimmy Kimmel just after they reunited in '08.