Do you actually need an ESRI "Geo-"database, or just a "spatial" database? Setting triggers and stored procedures on an ESRI Geodatabase is generally not recommended, as you may be trashing the geodatabase by your stored procedures and triggers interfering with the ESRI designed ones and data model.
If you just need a spatial database to store stuff, and want to manage it yourself, you can use the pyodbc Python package that is part of the package list in ArcGIS Pro in the Python Package Manager, to insert data in the database through cursors, and use database DDL and DML statements via ODBC to manage your database.
You can subsequently access the created data by running the Add Incrementing ID Field tool to add an objectid field (or manage a similar unique ID field yourself), and then use ArcGIS Query Layers to add the data to ArcGIS as read-only layers (that can however be input to other tools).
I have done all of this based on a PostGIS database installed on Ubuntu running in Oracle's Virtualbox virtualization environment on a Windows 10 host.
It takes time to get familiar with all of this, but is do-able.
Of course, since you are not going to create an ESRI Geodatabase through this method, editing data, and using any of the advanced Geodatabase behaviors and functionality, is out of the question. Be aware of that (Actually, editing data may be possible through a web Feature Service, but I have never tried that and it requires ArcGIS Enterprise / Server to publish the service, also see this link for information regarding accessing a non-geodatabase spatial database: Databases and ArcGIS—Help | Documentation ).