Performance after restart/reboot pre-Analyze

Hello,

It seems like whenever I restart my database server performance is quite bad until an Analyze has been performed. I expect this is due to the query planner needing new stats, but that leads me to three questions:

1. Is there a way to have Analyze stats/query planner "plans" persist through a restart? Like store them somehow after an Analyze and tell Postgres where to pull them from?

2. If not, is there a technical reason/limitation as to why?

3. Are there general configuration settings/optimizations for improving performance after a restart before an Analyze has finished?

If this would be better directed toward a different mailing list, let me know. I appreciate any guidance on this.
Also, if it affects anything, the environments are on AWS RDS, we're currently on 14 and 15 but will be upgrading to 16 very soon. I've only recently been brought into this, so I'm not fully aware of what kinds of queries we mostly handle or how the various tables/views are indexed.

Thanks,
Dave