Using Godot 3.5.2, if you attempt to export the project as an .apk file, and then use adb to install it on an Android device, what errors do you get?
I'm able to export to Android with both 3.5.2 and 4.0.2. I use the Android SDK provided in the current version of Android Studio (Android Studio Flamingo | 2022.2.1), which I installed using JetBrains Toobox. (You don't have to use JetBrains toolbox. I use it because I have multiple JetBrains applications to update.)
My OS is Pop!_OS 20.04, which is Ubuntu-based.