There is a new approach of using static site generators to make BI pages feel instantly responsive.
- Evidence.dev (https://evidence.dev/)
- Observable Framework (https://observablehq.com/framework/)
I have found that both the speed and the frontend control either of these tools gives you is pretty good (with Evidence looking better out-of-the-box just in my personal opinion).
The main problem I always had with the embedded versions of Tableau, Looker, etc. is that they felt super canned (it was obvious it was a poorly/"lightly" white labeled solution -- when you see an embedded Tableau dashboard, you _know_ it is Tableau, etc) and they were slow.
More on this here: https://magis.substack.com/p/an-observation-on-dashboard-spe...
PS -- I would add that the "headless" version of the above tools that I have seen is https://cube.dev/