Further update, I've downloaded the "other" Linux distribution from https://www.minecraft.net/en-us/download and it does create a .minecraft within /home/<USERNAME>/ and I have the same issue as with the Flatpak.
Failed to start download.
Name: 1.19.3-forge-44.1.0.jar
URL: https://piston-data.mojang.com/v1/objects/977727ec9ab8b4631e5c12839f064092f17663f8/client.jar
Error details: 0x0000000d - generic: Permission denied
Couldn't open file for download
Filename on disk: 1.19.3-forge-44.1.0.jar
Path: /home/<USERNAME>/.minecraft/versions/1.19.3-forge-44.1.0/1.19.3-forge-44.1.0.jar
Exists: Nonexistent