There are some tyres that are designed to wear through several layers of canvas before they are unservicable. I would not make a decision to accept an aeroplane with some canvas showing until I understood what type of aircraft, what type of tyres, what do the aircraft manual, the tyre manufacturer and the maintenance organisation say about it.
Andy, don't know who "the boss" is; could it be that he has sought information as above and reached the conclusion that the tyre is OK? In any case, the decision is for the aircraft commander alone to make so, if you're not happy then don't take it.
3 point