I honestly think that even if they had an unlimited tyre allocation, the system would still me a mess.
Let's say you just finished flying lap 1. The shortest amount of time you could have a better time posted would be 5 mins and 20 secs later on a typical circuit (like Bahrain).
That is after 1m45s in lap, 5s new tyres fitted, 2m00s outlap and 1m30s flying lap.
So you would have to be at least 4th from last when you finish lap 1 for there even to be any point in trying at all.
Say instead they raced on rocks that didn't degrade at all. Even then, they still would need the cool down lap to recharge the battery so even then, you are still looking at 3m15s before you can post your second time.
Instead, I think they have all realised that they are better off saving the tyre allocations and engine miles and not wasting time trying to do the impossible and make the system work. It can't work. It is an unworkable system.