It’s almost what you "don’t hear" that might sway you-- if you equipment is on a circuit shared with other outlets, fixtures or appliances, dedicated lines can help reduce overall system noise. I’m not talking about obvious electrical interference from other non-audio devices (more about that in a second), but what, for lack of a better term, may help with a lower noise floor: the ability to hear more against a quieter background. That’s the ideal, anyway. It comes from fresh wire, fresh receptacles, best practices in routing the cable in wall, among other things.
As to electrical interference, it doesn’t fully "isolate" from the rest of your household electrical system since it ties back to the same ground, per Code. I’m not an electrician or a Code specialist, but there are some here who really know this stuff and can probably explain some of the practices you ought have your electrician observe, starting with an examination of the present panel. (I assume you are in the States since I have no experience with audio and electrical wiring in other countries).
If you are already busting through walls, making a mess and hiring some electricians, why not install several lines- it will make life easier later, especially if you change or expand your system, run digital or video in the same room, etc.
I did my most recent install when we moved to Texas a couple years ago; hired a commercial electrician who pulled a permit, set up a new subpanel, used very heavy gauge feeder (4 gauge) to a subpanel adjacent to the room and 10 gauge to feed the outlets. I did opt to install a large (10kVa) isolation transformer system for the hi-fi.
I had four dedicated lines with multiple outlets installed in the listening room fed through the isolation transformer and a fifth fed from ’dirty power’ on the same floor for my air compressor and record cleaning stuff.
The results of this work may depend on the infrastructure where you are located: when i lived in NY metro, our electricity had issues. Here in Texas, with far newer infrastructure, even without the isolation transformer, the power was better.
My experience has been that electricians don’t necessarily talk ’audio’ and may even think some steps audiophiles take are unnecessary or just dumb. It pays to educate yourself a little here-- some of the folks I look to are @Almarg and @Jea48, among others.
You’ll also get some views from people who have gone to the trouble to get their power sorted and how that affected system performance.
I guess my short answer is: part of it is peace of mind to eliminate potential gremlins on the branch circuit you may be sharing, and old funky wiring and receptacles that should probably be replaced for better overall performance. That’s not voodoo. The other thing I’d recommend is interviewing a few electrical contractors to get a sense of their views, experience and knowledge. Having owned a number of homes over the years, and having gone through this exercise multiple times, the caliber of electricians I have used has varied widely.
You’ll want that permit pulled and approved by your local authority when the work is done. Not only for safety and Code compliance, but to avoid issues if you ever sell your residence.
I’m sure you’ll get more detail from others- this was intended to answer the more general question of benefit and multiple runs.