We'll see what happens with ETH 2.0, but ETH has had the exact same problem quite a few times. Anything with a fixed limit and variable fee is going to experience the same thing if enough users arrive.

And since ETH has stablecoins running on it as well as Dex's and no doubt plenty of other things the potential for pressure on fees is seriously high. There doesn't have to be a spike in price or new users to up it. The more services use it for contract type stuff the pricier it's going to get.