I think that the term site health might not loved by some people as they could associate this term to the beta site evaluation. If I understand correclty the process was changed some years ago. I think that it should be fine to use nowadays this term.
Personally I like the term site health because I associate it to voluntary personal health check ups rather than a evaluation imposed with veiled purposes.
Another name migth be "citizenship level". Honestly I had this idea tumbling in my head since a while, not sure why, but I know tht it's not original, see Replace accept rate with citizenship level.
2023 state
The Sites listing, as it shows sites statistics, Users pages and the Reputation League pages might help to have an idea about the site health but they need some tweaks.
Lets take a look to Meta Stack Exchange Site Statistics as of March 22, 2023 :

Missing Metrics
Some missing participation metrics that might be helpful
- Votes
Any kind of vote (Upvote, downvote, vote to close, vote to delete, etc.)
- Comments
- Edits
- Flags
- Active Chatrooms
- Active Users
By active users rather than using a reputaion score, I think that it might be more important to have being active in the site for certain ammount of dates during a time window, being active (post, comment, flag, etc.) during 10 days in last 30 days or something like that.
I'm not saying that they should be shown in the sites page, just that it might be helpful to make more eaisly available thant having to go to https://data.stackexchange.com. One way might be to have a site dashaboard page. On the sites page, it could be included a link to directly jump to it.
Missing Temporality
At this time it's not possible to see if the site health have changed over the time looking at the Sites listing as nowadays only shows "All Time" data. The Users pages have tabs to see week, month, quater and year data but it's not possible to compare with previous periods. The Reputation Leagues include features to see reputation changes, and see reputations from different periods (Curren, 2022, 2021, etc.) but it's not easy to see trends beyound the comparison with the previous period.
Data Explorer
There are queries from the data explorer that might help to have an idea but they do not really work as some attributes do not time temporal meta-data, i.e. it's not possible to compare how many active users were in month by month since the private beta to data explorer update date.
Posts with queries about the site activity, active users, etc.
Related feature-requests to "diminsional modeling"
Qualitative data
As it happens in personal checkups, the qualitative data is important but it should not be used as the unique element to take decisions. For now I will no go deep in the qualitive part other than to say that the a quantitative dashboard is helpful and the qualitive part might be continue be handled in the corresponding per site meta.
Changes
Ideas, some are changes to current pages, some are related to create a dashboard. This is not a feature request, I'm just sharing some thougts.
As mentioned above currently most of the aggregated data are site life time. It would be nice to be able to choose the period, i.e. week, month, quarter, year in the site statistic as could be done in the Reputation Leagues pages. This might imply a business intelligence approach ( use a different way to store the data, i.e. cubes), and might not be easy get historical data i.e. by mining the data dumps, but it will be nice that the Company start saving the new data in the proper structure / storage and in paralell work in adding data from the data dumps, probably this could be crowd sourced or might be handled through "connectors" / "plugins".
Modify the Sites listing, Users pages above to allow the user to view data of multiple previous periods, 3 as minimum, i.e. Year: 2022, 2021 and 2020. Monthly: March, February and January.
Add buttons / links to view the data as graphs to make it easier to see trends.
Ideally allow site moderators to customize a publicly available site dashboard by turning on/off predefined scorecards, tables or graphs to help promote participation according to the current site needs.
Users with access to the moderators tools should be able to see all the site scorecards, tables and graphs, including access to aggregated data, no details, about flags types and bans (post, review, etc.,) and users suspensions.