Sorry to hear of your misfortune. My only experience with Gieco was when one of their insured rear ended us in our car some years ago. My body shop (owned by a trusted friend who is an expert and has an ICar certified shop) had nothing but trouble with Geico's adjuster. He refused to authorize repairs that met ICar's standards so we eventually had to get my insurer, State Farm, to pay for the repairs and subrogate against Geico. Geico also refused to pay for a rental car for me for a period of time sufficient to get the repairs done...which were pretty extensive. Don't know how they are now but that was my experience with them.
Your experience reminds me to check with State Farm to ascertain how my policy covers damage related to a tire failure.