Actually Gene tried to get them signed.
Paul saw them first, then brought Gene down and he jumped right in with them. Recorded their demo's, signed them to his production company, brought them to New York to audition for Bill, their manager. The only one who wanted to move forward with signing VH and managing them was Gene. The rest of the organization including Paul turned them down. Paul has been very open that he didn't want the competition from them, he basically didn't want to give them a shot and get replaced by them. As he said, survival lol. In fact that and other things caused a rift between him and Roth for years.
Of course all this coincides with Gene in talks to work with George Lynch's band, the night Gene showed at the Starwood, VH was opening for Lynch's band. Gene went straight for VH. Probably for the best for Van Halen, they would be signed I think around a year later to Warner Brothers with the help of Marshall Berle who later managed Ratt and Ted.