My suggestion, though, is different. If NIST has such an approval process, then, Apple/Google/Mozilla/anyone else doesn't have to go through an approval process, but if they do go through such process then they can put on their product, the sign "Approved by NIST as web browser compliance".
However, I am not so sure that such a thing is even necessary at all. I think the use of the web browser should be reduced in order that this is not necessary.
By an organization for use by the public at large? Yes!!
Devs can still make browsers explicitly for other devs or for non-public use. If society depends on your product to function then your product must be regulated. Imagine google auto-updates chrome to stop supporting http or something, imagine the economic chaos. The whims of Googlers are not something the public should rely on. Everything from their standards compliance to their change control should be regulated,same goes for mozilla.
Perhaps stanrdizing the QA process would have prevented this? The problem I was addressing is this issue falling inline with a pattern set by google. They wouldn't test only against Google if Google's whimsy divergences didn't mean it cost too much to support anyone elss in your QA.
Maybe I'm underestimating Twitters developers, but in this case I honestly doubt Twitter had a QA process testing if things remained in browser cache at all, until someone pointed out that there is a problem. If they had thought about it as a problem, adding the cache header would have been easier to implement and test for than browser testing somehow.