We also had an issue with HL and New Quizzes just a couple of weeks ago. Some students were getting an error message and some were actually able to take it but the sessions weren't being properly proctored. When one student reported an issue directly to HL, they were told it was an issue with the instructor's quiz settings.
When I reached out to HL directly, I was told "Unfortunately, there seems to be a bug with the New Quizzes that is currently under investigation." They said that they would add me to their ticket and would let me know once the issue had been resolved, but I have yet to hear anything.
So I've been suggesting to folks that they continue to use the "Register Canvas New Quizzes as Third-Party Exams" option until we get confirmation that the bugs with HL are fixed.
Will definitely appreciate hearing about it if it's working well now for other folks.