ETH (which was commonly thought to be true) implies P≠NP.

sure, this proof itself doesn't settle the P/NP problem, but we've lost one of our core beliefs.

I think that statement is too strong -- if ETH was a core belief then its implication P≠NP must also have been a core belief ... but I don't think that those who assumed ETH held it that strongly. As you say, it was commonly thought to be true ... as was P≠NP ... and the latter still is. But is not a "core belief" -- I think virtually everyone believes that they could be wrong, whichever side they're on.