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Why does Ethereum not use a second price auction for setting transaction fees?

This would protect users from paying too much by mistake and lead to more stable and predictable fees for everyone.

Implementing this for Bitcoin is discussed in the paper Towards a Functional Fee Market for Cryptocurrencies

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Such a proposal was raised on ethresear.ch in July 2018, and is still under active discussion.

https://ethresear.ch/t/first-and-second-price-auctions-and-improved-transaction-fee-markets/2410

Why isn't this already implemented?

Firstly, it'd require a hard fork. (No big deal; it could go in with the next scheduled network upgrade.)

Secondly, you're changing the incentive system and therefore the cryptoeconomics at play. The current system works and is well understood from a game theory perspective. Changing things would presumably first require drawn-out academic discussion on how best to design an implementation. We're just not there yet.

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