I've been making chicken broth pretty much the same way for decades. I cut up a whole chicken, cutting through the bones of all but the breast and thighs (to release marrow). I pretty deeply brown the pieces (with skin) in the bottom of a stock pot, then remove. I then saute roughly chopped onion, celery and carrot in the chicken fat until browned, then I deglaze with a little wine or sherry. The chicken pieces go back in the pot with a gallon of water, I bring just to a boil, cover and turn off the heat. Now it sits for a half an hour or so. I then fish out the breasts, refrigerate the breast meat, and return the bones and the skin to the pot. I simmer the remaining pieces for about another 40 minutes, remove the thighs, reserve the thigh meat, break the bones, return the skin and bones to the pot and slow simmer everything for the better part of the rest of the day. I allow the broth to cool, strain through a colander then a fine sieve, then refrigerate overnight.
Here's what's weird. Half the time, I can simply remove hard solid fat from the top of the chilled broth. The other half of the time the fat separates and thickens, but never solidifies, I end up dirtying my fat separator. I can't figure what could possibly be different from one batch to the next. I generally even buy the same brand of chicken and I only add a tiny bit of oil (vegetable) for the initial browning.
Even weirder, sometimes the broth is so full of gelatin that it is actually harder than the fat, it makes for a good soup when that happens, but makes separating the fat a bit of a pain.