If you have an issue with using the add-on, please report a bug.
Simply importing the PSA into the scene will not automatically apply the action to the armature. This is in part because a PSA can have multiple sequences imported from it, and also that it's generally bad form for importers to modify the scene when they don't need to.
The PSA importer creates Actions for each of the selected sequences in the PSA. These actions can be applied to your armature via the Action Editor or NLA Editor.
Unreal PSK/PSA (.psk/.psa)
The add-on can't import the psk model properly and it also can't import psa, leaving a false imported successfully message in the logs.