Sara_Cummings Haven't commented myself before, but we are also suffering from the issue with custom templates being hidden behind "Saved on this site". We have exactly three templates that get pushed to all our sites by a power automate flow, so depending on where you draw the line it may or may not be an acceptable solution for us to automatically show that tab first where there are "many" templates.
Another option that would be even better for us is if you could provide a way for us to add custom, tenant level templates, that would show up as the first option on all sites. We have 100-something sites that we consider our core intranet, but since news posting can be done on any modern Sharepoint site we have a requirement to have our custom templates available on all the 40000+ sites in our tenant. As mentioned, this is managed today with a flow that gets triggered on site creation, but needless to say updates to templates are cumbersome to roll out. It would be a great improvement and simplification if we could add them tenant level instead. Any hopes for such a solution? 🙂