I remade the model but this time i applied all modifiers before rigging
(this new model also looks a little bit different, but that's not important for this question).
The new model (Blend file attached) is made with a Three-Step process, with each step being saved as a Collection in the Blend file:
Low-Poly Model
This is the original Low-Poly model with the Modifiers, for easy editing.
I also made the Skeleton in this step, but it's only for visual reference yet.
High-Poly Model
The modifiers are applied.
Rigged
The High-Poly Model is rigged using the Skeleton.
With this adjusted workflow, the new model can actually be exported into Godot
as an animated model instead of being stuck in the T-Pose.
I also decided to test using the old model again and it also animates as expected.
I just didn't know how to test the functionality of the Skeleton in Godot before.
Now i know that i Can continue using this workflow for animated models
because i could confirm that the exported model can be animated by Godot.
@Megalomaniak @fire7side @Gowydot Thanks for the hints.