Confused about Entity bundles
After giving you this context, my actual question is, if I will only
ever need 1 type of every entity bundle, do I really need entity
bundles? What would entity bundles do for me?

No, you don't need bundles. Configurable fields are stored in bundles. Since you don't want to configure fields you don't need bundles.

I still wonder, if I will have only one bundle per entity type, do I
need to set up a configuration entity for the the bundle?

The core user entity is an example for a bundleless entity type for which you can configure bundle fields. It looks like having one bundle but there is no bundle config entity. The bundle fields use the entity type id as placeholder.
So even if you later need to add configurable fields, through UI or YAML, you don't need to set up a configuration entity for the bundle. See Adding Field To Bundle-less Entity On Installation Not Working