Discussion about the Korg PolySix synthesizer

First things first:

The heat "problem" usually is not a real problem. The back of the Polysix
doubles as a heat sink for the power supply.

Now, the LFO problem.
The Polysix is insofar a bit different than most synths as the "VCA"
modulation is not really generated in the "real" VCA section of the voice
board. It's generated on the effects board instead. This means that the LFO
signal is passing from KLM367 to the effects board together with the
effects mode and effects intensity signals via CN03 (the orange 7 pin cable
on the left side of KLM367). This connector is dangerously close to the
battery and I've witnessed corrosion in this area frequently. So I wouldn't
rule out the possibility that something there is going wrong.
You may want to check if the symptoms depend also on the effects type.

Last but not least: The problem could also come from the effects board. I
have seen several effects boards which had also battery damage on the edge
next to KLM367.