The phono input is more sensitive and also has RIAA equalization. Turntables are the only thing that should be connected to that input.
Due to the fact that you say it sounds OK through the tape inputs but not Aux, and the output is louder in one channel, it's likely not a sensitivity issue but dirty connections and controls. Clean those first. Give special attention to the Function switch, and note that the majority of the switching on that control is done with sliders on the board, rather than rotary switching on the shaft.
Thanks, I was waiting on a new Bluetooth adaptor which finally arrived today and it pretty much confirms your info regarding input sensitivities. Up to now I’ve been using a small battery-powered Bluetooth adaptor I had to hand for my car, which is probably very low output (hence working better in Phono than Aux). The new (mains powered) Logitech one arrived and works great with Aux (but it was too much for Phono… I tried it very carefully to test the theory). They probably all need cleaning too, but the main issue here was understanding the different sensitivity levels of the inputs and pairing the right modern equipment with it. The cheapie Bluetooth adaptor can go back to the car where it works fine.
Will investigate the L/R issue, thanks for the pointers. Have been short on time as my father’s Arcam Alpha 2 suddenly stopped working and has been fun to fix (definitely at the other end of complexity and size to the 7065!)

