Might not even be the volume controls, could be the internal record bar, which is a long bar looking switch. All the signals go through this switch. Sometimes, when the connection is bad, cranking up the volume control could increase signal voltage enough to overcome the poor contacts sufficiently to restore volume for a bit. As a test, I suggest you put a tape in the deck and press record (might need to depress play at same time). Then stop. Do that for like 20 ~ 50 times. If you are getting improved results, it's virtually a sure sign that it needs cleaning. Squirting some deoxidizing solvent cleaner in there and working it will restore it. Or it could the the function switch. If you want more detailed information on how to do that, just search the archives. This question "cleaning the record bar" has been asked and answered countless times.
As for the amplifier, I doubt that it's the amp at all. The very fact that you are getting output, and loud output (albeit intermittently) is proof that the amp is capable of outputting a strong signal. The problem is that you need to feed the amp with a good signal to get a good signal out. Garbage in, Garbage out.