That's part true since the FBO is the insured, not the OP, however to say the claim would be denied because of the acts of his employee is incorrect, that is one of the things he insures against. If you crash a rental plane, you never get denied a claim from the FBO's insurance because you aren't the insured party, the FBO is.
The insurance has the option to subrogate the claim against the OP which means in order to get back the money they have to sue him.
If he has renters insurance, they will subro it back to the FBO because they allowed him to take up a plane lacking qualifications. With that log book like that, if the FBO handed him keys to solo, they hold all the liability.