Is there a list of all games that support the 'new' gamertag system?

Not all games support the new gamertag system. Since yours has special characters, games that don't support the new system will show your old gamertag.

I thought games that don't support the new gamertag system use the "Xbox games that don't support the new gamertag system* and all Xbox 360 experiences" gamertag? So are you saying a game that doesn't support the new gamertag system can still use the "Xbox one console and Xbox apps" gamertag, but not support the suffix styling?

Not all games have been updated with the new system. So you will see that in some games.

https://support.xbox.com/my-account/manage-gamertag-and-profile/gamertag-updates-faq

Does the suffix always appear next to my gamertag?

If you’re assigned a suffix, the # symbol and suffix appear everywhere except in friends lists. You don’t see suffixes of friends in your friends list and they don’t see yours in their friends lists. Your suffix
is used in games so that everybody knows who’s who.

When will games start supporting the new format?

A few games released this year might support the new format, but most games won’t support it until 2020 and beyond. It may take a few years to migrate all new games over to the new gamertag format.

Yes. Since not all games support the new gamertag system (which includes those characters), those games will show your old gamertag.

"On Xbox games that don't support the new gamertag system"

This is what I'm talking about. We're two years in and no games support the new gamertag system.
Genuinely infuriating.