"Databases are usually locked down to a few team members..."
Not true in my experience. True at very large / very mature companies, but at any In that same vein, in a small company environment you don't harass engineers with query requests, you learn SQL. I've seen hundreds of non-technical users do this (and helped many of them), it's not unusual. Visual query-building tools never help. Even good interfaces like Looker, on top of carefully-crafted data warehouses (which most won't have) are still vastly inferior to SQL and so analysts don't use them. Again, just trying to suggest the problem may not be what you think.
Generating queries with a language model won't work. Real data is far messier than you expect. The table names and field names will not be as literal as you need them to be, and half the database will come with "special instructions" (like "oh you need to divide that field by 100 because...") that will not be inferrable from the names or the data. Many DBs will completely lack foreign keys or consistent key naming. There will often be epochal "eras" in the data where everything before YYYY-MM-DD worked like X, and after that it's Y, and the value of that date is recorded nowhere.
"Currently we support Postgres, MySQL, Snowflake, BigQuery, and Redshift."
I know Java isn't hip but consider relying on JDBC for DB access. There are more DBs out there than you can imagine, and 100% of them support JDBC. Otherwise there will always be some DB you're not supporting, and adding that support will be costly.
Anyway just some food for thought. Good luck. (about me: 18 years in data & analytics)