My suspect initially would be either a stuck or liquid damaged button,
or a mechanism that is out of alignment in the vcr.

Since you have a power indicator working, the power supply must be
operational at least in the 5VDC. As the DVD/VCR button changes the
selection, most of the power supply again should be working.

If you get the unit apart it is a simple matter to check the power
supply voltages. You should measure 5VDC at any of the buttons to
ground and around 12VDC usually going to the motor drive circuits (may
be 6VDC or 9VDC).

You will not be able to diagnose any kind of mechanism faults without
the service manual and you will likely not be happy with the price if
it even is available, call the number in the back of the owners manual
for the brand.

As the buttons do not seem to work, you have no way to activate the
tv/vcr button to turn on the output for channel 3 or 4.

Also before doing anything at all, make certain that the unit is not in
some kind of child lock out mode. Again reference the owners manual.
Most modern vcrs and tv sets have a child lock out mode.