Maybe a few other sites (each site is of course centralised but as a whole you have many points of failure) which offer "trading tokens" or something in exchange for real money.

A token purchased at one of these sites should be fully redeemable at any other site, so if any of them decide to shut up shop at any point there are more you can use.

Then you deposit this token at the P2P exchange and it acts like fiat to buy/sell btc with.

How would you secure this, make sure the tokens are valid, can't just be made up, sold/used twice, do money laundering checks etc etc? Hell if I know, but this is the sort of thing I'd like to see. Separate fiat from the BTC exchange, make it someone elses problem.