As for the Postgres-derivates that you mentioned (Timescale, Citus), we have not tested OtterTune for them yet. But there is no reason they shouldn't also work because they expose the same metrics API (pg_stat_database) and config knobs. There are just way more people using Postgres (RDS, Aurora), so we are focusing on them right now.
First of all, congrats for the project.
As you probably know, you and any other user can indeed manage Postgres configurations, in any environment, via an API. See the "MANAGE" section in https://postgresqlco.nf/ Essentially, a hosted "Git repository" for Postgres configurations.
As I said above, everyone that we've talked to rolls their conf management tools because they are doing more than just DBMS confs (proxies, networking, kernel params, middleware, etc). Lots of Terraform.
Yes, indeed we have considered doing a TF module for the configuration management API :)
FWIU, Trove supports 10+ databases including MySQL and PostgreSQL.
Here's the official Ansible role for Trove, which provisions various e.g. SQL databases on an OpenStack cloud: https://github.com/openstack/openstack-ansible-os_trove