You're very likely going to need a thickener, especially if you don't make your own stock. (On a side note - I highly recommend making your own stock at least once. You end up with a vastly different stock/broth when you do, and in this case, it might achieve the consistency you're looking for, because a good animal based stock has gelatin in it already.)
Gelatin shouldn't generate "blocks or blobs" when incorporated correctly, but if you don't want to use gelatin, you can also add corn starch, arrowroot powder, potato starch, tapioca flour, or other starch to thicken the broth. (Flour is an option, too, but I prefer other thickeners, since in my experience, it often takes more flour to thicken something, especially later in a cooking, and the dish takes on some of the flour's flavor and texture.)
Gums (the compounds often used to give non-dairy milks their creamier qualities) can also work. Xanthan gum is pretty easy to come by, though you may also be able to find guar gum. These usually only require fractions of a teaspoon, so shouldn't be enough to affect taste or most of the other properties of the broth except thickness.
For all of these, the usual way to incorporate them correctly, so they don't create blobs, is to thoroughly dissolve them in some water before adding them to the cooking pot. You'll want to use a whisk or similar utensil, to ensure any blobs are broken up.
If you want something more akin to stew, as opposed to gravy, you should only need to cook down the liquid in the broth for a while, or just drain off some of the broth when serving. The increases the ratio of vegetables and meat to liquid, which gives the dish as a whole a heartier and more filling feel, without actually making any of the components thicker. This is essentially how tomato sauce vs paste works -- the sauce has a higher water content than the paste. Because the tomato puree doesn't dissolve in the water, the result is a thicker or thinner solution. You can puree the ingredients in the soup (vegetables, meat, etc) to get a tomato sauce/paste kind of consistency.
What ever it is, sounds like sinew or other hard tissue, it's done it's purpose now the flavour, sugars, geletine etc is now in your broth and you should probably discard them. I don't imagine they will taste of anything now and the texture is likely to be vile.
By all means taste a bit to see what I mean. It won't make you ill though you may possibly gag...
Best Answer
Yes, you can. The advantage, as you point out, is that you can collect the bones until you have enough to make a flavorful stock. They freeze well, but will last even longer (avoiding freezer burn) if you can remove as much air as possible from the package. I often do with with chicken bones, necks, and skin until I accumulate a decent amount for a stock.