I see you mean this (the plunger and the part inside). Is there a way of doing it without unsoldering the solenoid? I am really not that good at soldering.
Also if this was the problem should I hear the relay clicking and then the arm lifting with some delay afterwards?
What I think happens instead is that the relay triggers with some delay after the botton is clicked or whenever the arm gets to the end of the record. Added a video for more context