Welcome to Apug!
That the button does not work and the cable release well, ist not that surprising seen their design.
The classic design is that the button is a shell, with a release thread, that takes inside a knob. This knob can be pushed out of that shell by means of the cable release plunger, by this activating the shutter without pressing the button. The button just acts as counter force to the plunger. So that knob is independant of the button. If the button is pushed however it soon after contacts the knob and pushes it further to activate the shutter.
I very often come across cameras with sticky substances in recesses, slits etc. Typically that seems dried concentrate of fruit juice, cola or such from splashing a beverage on the camera. In your case it may have been someone with marmelade fingers...
Yes, using a solvent would be an idea. Seen the above I though would not use alcohol but instead a most tiny amount of window cleaner. If that does not free the shell you can try alcohol. Next step would be taking off the top cover and taking out the button from the cover. It also may be a tiny grain of sand making the button stick.