I strongly disagree. The Casimir effect's math works, yes, which means that "there exists a sense in which the summation equals -1/12 in this calculation", but I interpret this to mean that the calculation being done isn't the most accurate representation for the physics at hand. QFT is, after all, filled with weird tricks involving infinities - it seems perfectly plausible to me that the Casimir effect's math actually involves something like "-1/12 + <infinity terms>", but we're only measuring the non-infinite term in some sense, so we get away by only using it in our calculations.

I think it's Very Not Good to see a totally unplausible mathematical result in physics and take it as anything other than an open problem that needs more work. It's okay to be amazed by it, but it's not okay to say "sure, okay, let's just leave this as it is".