He’s Not really correct. I’ve had huge improvements going to a dedicated circuit when OTHER ITEMS on that same circuit are noisy. Fluorescent lights, hair dryers, microwaves and vacuum cleaners are quite noisy and will cause visual (TV-projector) mouse as well as audible noise. A dedicated circuit can help in those cases.
A good wiring technique (USA) is to put the noisy items on one “leg” in the electrical panel and keep the audio items on the other “leg”. (In the USA, boxes are fed with dual 120V wires or “legs”; most circuits only use one, unless they are a 220V circuit. Air conditioners, stoves, air compressors, welders are 220V and all else is 110). The wiring strategy is to keep bathroom circuits (hair dryers) and living room / bedroom / hallway circuits (vacuum cleaners) and buzzy lighting circuits (fluorescent) on one leg and audio on the other.
His assumption in needing to run a new ground is based on issues with ground loops. If the length of wiring on all audio circuits to the box is the same, ground loops are often avoided until you introduce coaxial cable into the mix. Use isolators if that happens. RCA cables (subwoofers) also cause ground loops; in that case consider higher-end gear with XLR interconnects or use isolators on the sub RCA’s.