I have never heard of not putting cooked (vs raw) food into compost and am at somewhat of a loss to think why one might say that?
Really you can compost ANYTHING YOU LIKE, as long as you are willing to live with the results. The issues to consider are mainly:
1) some things attract rats/dogs/coons more than other things. This is arguably the biggest knock against meat, bones, eggs and oils. A burgeoning rat population is NO FUN and can be awful hard to do anything about sometimes; and dogs or raccoons can scatter your pile alllllll over. Burying the stuff deep and mixing it in with the existing compost helps. In particular, if you're going to compost eggs I strongly (from experience with rats) suggest breaking them and mixing them into a large volume of compost pile, rather than creating a nice little stockpile just waiting for a hungry rodent.
2) high-protein animal matter can encourage growth of food-poisoning type microorganisms that may still be there in some numbers when you put the compost on your garden. Whether and how much you worry about this is goign to vary a lot among people.
3) too much oils (like, emptying out your turkey deepfryer) can sog out a pile and cause poor and mostly-anaerobic composting with a poor result.
4) and, obviously, if you use anything containing herbicides or other pesticides that won't break down very rapidly, you'll end up with some amount of that in your finished compost and may have to live with the results.
Personally, I don't compost meat products or most table scraps, b/c the chickens eat them and because I don't want to make rats' lives any easier. I sometimes put small bones in the compost pile, but most large bones go to municipal composting (we have green-bin pickup here). I never HAVE lots of kitchen oils to dispose of, nor anything with poisonous chemicals in them, but if I did I would not put large amounts into the compost.
One thing to consider -- if you have a decent-sized property, it is quite possible to have multiple TYPES of compost piles, and one of them can be stuff that you are really just basically trying to get rid of rather than trying to put on your garden per se. I chuck that sort of stuff in a low windrow where the neighbor's field overflows into our backyard when it rains -- I figure anything I can do to slightly raise the level of the ground there to slow down the floodwaters is all good
I will not use municipal compost for anything at all around here anymore, even tho you can get a certain amount free each year. It LOOKS great, but I have gotten so many Very Exciting Weed Seeds that way, I'm just not doing it anymore. Plus as others have said you have no idea what the chemistry of it is. If I were very very hard-up for soil I might use some of it but only at the BOTTOM of my beds, covered with something safer.
Good luck, have fun,
Pat