Definitely. I poured out the water and was left behind was all the white tallow with all the chunks of cooked meat embedded. I then melted the whole lot and the chunks were then easily removed. Maybe that's the way others do it but it seems like an extra step.
What I always do is this:
I take a pint of old tallow and I melt it in my stock pot on simmer. Once it's completely melted I start adding in chunks of new fat. Starting off with a bit of tallow is really helpful to get the process started.
Then we start carving our carcasses, taking steaks, removing roasts and stew meat, and cubing meat for the grinder. Any dirty bits, silverskin, nasty bubbly fat, gristle, bloodied areas, gross stuff, etc, just gets thrown into the stock pot. I keep adding and adding fat like this until it's at the very top, and it simmers all day long. At the end of the day I pull off all the junk that is floating and turn off the simmer. In the morning it's cooled but still not hardened, so I simmer it again to bring it back to temp and continue the process.
It takes us several days to completely process a carcass so the pot stays on the stove during the whole time. When we're all finished with the meat we usually have the stock pot at the very very top full of tallow ready to be skimmed, strained and finished.
