Peat has no nutrients whatsoever so it is simply a Carbon source, a "brown". However, peat is also something the bacteria that digest the material in your compost pile do not particularly like that and will avoid it unless nothing else is there to eat.

You don't have to throw away "spent" potting soil. You can rejuvenate it with compost additions and other organic additives. I have a 5x5x5 soil bin outside. Every year when I dump the pots into it, I toss in some manure, some compost, some seaweed, some chipped leaves, whatever, after every foot or so of potting soil. I use old potting soil every year, some of it that I purchased in the early 90s. Works fine. Sometimes I bury a hunk of fish in the middle of the pot. I do that for my potted green peppers, who appear to get a kick out of it.

In my opinion peat and FINISHED compost are the same thing with one slight difference. Peat has been sitting in water for 1,000 years so most of the microbes that might live on it are water types. Furthermore most peat has been 'processed' with heat so it is most likely pretty free of microbes except what it picks up from the air. I don't even think of peat as a brown simply because it finished composting about 999 years ago.