Like caution pointed out the solenoid when it activates, has to engage some part on the gear and lift the head assembly. Manually raise the head and see what all sections are involved in that movement. See what is hanging around the mode gear and learn how it facilitate the movement of the play head mechanism. Do you see a missing arm/follower that engage with the gear and the head assembly movement? It has to be a cam and follower mechanism. We have to find the follower here, or both and how it is connected to the solenoid. I understand it is easy to say all this and in reality, it is very hard to locate all this.
I have worked a lot on cassette decks, and some are so complicated. Fixed many and was unsuccessful in some cases. I have a like new AKAI GXC735D that I am working on also has the same problem with the play function activated by soft touch buttons. I was unable to find the cam and follower! I lost lots of hair over the years and decided not to fool around with after trying more than 25 times working on it.