you can add some chunks of carrots/celeriac etc with the carcass to make the stock for added flavour. Just make sure you don't boil it, but simmer very gently when it comes towards the boil (prevents lots of scum forming].
I guess I always make the stock first, let it cool and then skim off the fat that congeals in a nice & easy to remove layer, and also sieve it to remove any small bits of bone and gristle that may be present (that's what all the clever books say anyway). There is normally a bit more meat that can be taken off the carcass and added back into the soup later [or be put aside for a risotto - that's what I do with most of my chicken stock, you definitely want to remove the fat for that!]
But back to the soup:
Fry onion/garlic and any other veg briefly before adding the stock ... tastes better than just chucked raw into the stock, especially the onion/garlic!
But your method should work fine, if the carcass isn't too fatty, it just seems a bit "backwards" compared with the conventional way.
Whatever you end up doing, let us know how it went
T