My $0.02:
Yes, doing a fresh install and then connecting and upgrading your old database is a feasible approach.
It mostly comes down to making a fresh install "by the book" (including AWI/Tomcat, the new JWP process, and (optionally) Analytics/Postgres), and then updating the database with the db load utility.
You should, however, pay close attention to the individual change notes for all the intermediate releases: there are things that may need to be thought of in your UC4 objects, like changes in the scripting language etc.
So I suggest to start by carefully reading and following the update instructions from 9.x to 10.x, 10.x to 11.0, 11.0 to 11.2 and so forth. And when you do update your database, of course make sure you have backups of the old state, and better yet, work with a copy of the db.
As for any update, perform this in a test environment first, document all your steps, use the test variant for some weeks before finally upgrading production.
Best,
Carsten