Also, I know it's nitpicking, but does it bother anyone else that Pike and his crew are wearing TOS uniforms instead of the much simpler and muted 'The Cage' uniforms?
Not at all, because those uniforms wouldn't look any good in a modern TV show, which is what this is. Roddenberry himself considered
Star Trek essentially a dramatization of the "actual" events, just the best approximation of the future that 1960s Desilu/Paramount resources were capable of (within the limits of time constraints and executive meddling), and he didn't hesitate to replace it with a better-looking approximation when he had the means and budget to do so, which is why virtually everything looked different in TMP. This is not real life, it is art, and that means it can be changed and improved.
It's the same thinking behind the Kelvin Timeline uniforms and the design of phasers in both Kelvin and DSC. To the vast majority of the TV audience, the iconic 23rd-century uniforms are the bright, black-collared ones from TOS, while the pilot turtlenecks are an obscure footnote at best. The goal of the costume and prop designers is to evoke the more familiar, iconic designs, the ones that resonate better with the mass audience and frankly look better.
Besides, even if you stick to a strictly in-universe perpsective, Starfleet uniforms change with such absurd frequency and randomness sometimes that it's not that implausible that the turtleneck uniform style could've been in use in 2254, been dropped by 2256, and then had a brief comeback by 2265.
Though those uniforms were, I believe, used on Kirk's enterprise - that's 'TV canon'.
"Canon" does not mean every last detail. The canon is the overall, collective body of work, and any ongoing canon changes its details over time, which is why we don't talk about the adventures of James R. Kirk and his part-Vulcanian first officer aboard the lithium crystal-powered UESPA vessel
Enterprise.