There is no connection between Ethereum and the stablecoins on the basis of them keeping ETH alive. The Ethereum blockchain is just one of the blockchains in which those stablecoins run and without those stablecoins, ETH will exist. Likewise without the Ethereum blockchain, stablecoins will exist. Now we have more chains such as TON, TRX and others where more stablecoins are traded than ETH, yet ETH is still doing so well.
That's true that ETH will keep exist but I think if stablecoin doesn't exist in ETH blockchain, the TVL will only be fraction of what we have currently.
People will definitely grow hesitant to stake and provide liquid to various defi since there will be no sure way to mantain against volatility except hedging on cex with equal amount of money to short to mantain delta neutral.
However, the fact that most of people in defi are locking stablecoin speaks something about how important the stablecoins are.