John and Richard, if you look at vinyl carefully, it looks as if the disc is considerably thicker at the label, then flat until it reaches the 'start' edge. I don't think that the 'glass idea would work perfectly unless there were 'label-sized' holes in it, or perhaps foamed plastic (with holes) on either side, which would then mean you needed more heat because it would be insulating the record....
Bugger it! put a little weight (Bluetac? Plasticine?) on the tone-arm and digitise it.
Liz, you could download a shareware(limited)copy of LP Recorder and try that for your tapes. It doesn't seem to mind what the input is, and is simple to use. Or I couldn't use it! It is just a digitiser. If you wanted to clean up, there is Audacity.
Might work, and it costs nothing. Works with my ordinary tapedeck, piggybacked into the line-in with turntable (which HAS a preamp) and a minidisc recorder.