There may be some sort of "compromise" material that could be developed that would lack the toxicity or microplastics issues without compromising safety, but I'm going to defer to experts in that field.
Of course then you'll have a lot of lower-income folks driving around on dangerously worn-out tires because they can't afford new ones.
By the way I thought it was the synthetic rubber tires that lasted longer, but certainly possible that I'm mistaken.
Once the petrochemical supply is exhausted, or what is left is extremely expensive to extract, they won't be able to afford new tires either. (Tires, and personal passenger vehicles, could be some of the least important of the "missing" consumption from the loss of petrochemicals.)