Thanks for the pics. The 'works with max volume' seems to me to indicate that there is enough power (at max) to overcome a resistance problem, however briefly.
The cable being bent is no problem, as long as it is terminated correctly at both ends it makes no difference. It does look like it is one of the channel power supply cables though, given there is an identical cable at the other end of the connector.
What caught my eye though is what I hope is either a Speaker protection relay (or an anti 'thump' relay) just to the left of the 'green circled cable. It is in a clear plastic case. Next to the big capacitors which I hope were the ones that you were pressing originally.
If it is a relay, with changeover contacts, I'm wondering if (with the power removed - can't tell you how many times I caused more problems by investigating with the power on) you can remove the cover and clean the contacts. If you don't have a contact cleaner spray, just fold a piece of paper a few times to give it strength and then slip it between the contacts a few times. Also ensure that there is enough travel when the contacts make (either way when the armature is operated (pushed in) and released) the contacts should have follow through and follow back.
If that is not the problem, and you can still turn on the right channel by pressing on the board, then my other suggestion is that you power up the amp (yes I know) and with the plastic handle of a screwdriver or any other non metallic probe start gently tapping components and see if the right channels comes on. Without a schematic you could trace the circuitry but you don't know if it is a 'power'problem or an 'audio' one. There isn't any 'hum' in the speaker by any chance, indicating that the power is there but not the audio?
It is still looking like an intermittent open circuit problem which is usually either something loose or broken or not connected properly. Electronic components can go high resistance and then things don't work but it is very unusual that they restore back to normal.
Again apologies if you know a lot of what I've said.
Thanks to OldTurkey03 we now have a circuit diagram. See his 1st comment above.
Initially I suggest that you visually inspect the right channel path, starting at the right speaker + terminal on TM401 R+ connector point and working back thru headphones connector jack CNJ401 contact then to relay RY401 contact, R468, R463 and onto pin13 of IC401 on the Power IC board. (pin 1 is usually marked with a 'dot' on the IC case, then you count down that side then back up the other side to opposite where you started).
The volume control is a double potentiometer, one for each channel. You may also like to inspect from IC401 pin18, C452,R452, to volume pot RV401/2. This is the input path from the volume control.
After all this it may be that OT03 is right when he suggests that the Power capacitors may be failing. If you have a Voltmeter check that the voltages marked on the circuit for IC401 are correct e.g. -ve 31V on pin14. Just be careful when you measure that your Voltmeter probe doesn't slip and bridge 2 points, You might be looking at more problems. Also be careful yourself.