If you haven't gotten the Google Rewards app, your missing out. Free play points for simple quick surveys. I get so many books and movies for free.
https://play.google.com/store/apps/details?id=com.google.android.apps.paidtasks
Yikes! I'm sorry to hear this, for what it's worth. I went from a Pixel 1 (which I loved and used for four years - I still have it although it is noticeably slow compared to new phones) to a Samsung Galaxy Note 10+, which I loved the hardware but hated the software, and then to this Pixel 6 Pro, so I can't speak as to having experience with the Pixel 4 specifically.
However, if and only if you are or would consider rooting for the annoying "Internet" toggle, there are many solutions. I'm currently using Better Internet Tiles which requires root. It has the side advantage of having an optional toggle to keep your Wi-Fi and Mobile Data toggles from working from the lockscreen, if you choose.
As to the late GMail notifications, this was one of the reasons I hated the Samsung software. I would get work emails (we use Google for everything at work) anywhere from 45 minutes to two hours late. Some emails I would get the notification for after I already saw the email in Chrome on my PC, and had deleted it. So, it was even worse. I also had (and still sometimes have) problems with getting Google Chat notifications in a timely manner. My boss uses Chat a ton to tell me important things, unfortunately. Again, I have at least a much better experience on the Pixel 6 Pro than on my Samsung, and in both cases, this is after utilizing Android's built-in methods to keep these particular apps from being ignored/put to sleep.
As you indicate, they need to do much better. I hate to suggest it, but it might be worth your while to factory reset after having verified that you have everything backed up from your internal storage and Google Photos (free space in the app, and then verify there are no photos or videos left on the internal storage that you wanted backed up). Sorry I don't have any better suggestion at this time.
Good luck!
Maybe my success through X-Plore was related to the fact that I have my external drive formatted with ExFAT, and the Google Pixels don't include ExFAT support, so I finally paid for exFAT/NTFS for USB by Paragon Software. I still accessed the data through X-Plore, just mounted through the Paragon app. Hopefully, the late January update fixes the issue for us, too.
This may be done automatically, but heres a way to manually do this! 1. Get the Google one app ( https://play.google.com/store/apps/details?id=com.google.android.apps.subscriptions.red ) 2. Open it and scroll until you see the option to back stuff up 3. Follow the guide and backup your messages and more!