Should I keep the meat boiled off of bones when making beef broth?

Taste the meat and if it still seems edible to you then there is no reason to throw it out.

When I make stock, I keep it on a simmer for much longer than three hours and any meat is completely tasteless by the time I'm done. Three hours, however, is about how long you would cook meat to make a stew, so it's quite possible that you could eat it.

On the other hand, not all cuts of meat stew well and I'm not familiar with this cut. If the meat has plenty of connective tissues (i.e. collagen) to gelatinise it shouldn't dry out too much. But the proof is in the pudding: taste it.

I never throw it out. There are plenty of applications for mostly-tasteless meat shreds.

I mostly make chicken stock but I will use the meat shreds in soup, pot pie, casseroles, etc.

With a strongly-flavored sauce, you won't mind the meat not having much flavor. The protein value of the meat is still intact, so it is worth using it up to stretch the family food budget. Curries are a good use for soup meat, and as Sobachatina said, pot pie. I make a thick onion gravy as a pot pie base and that makes even the most tasteless soup meat go down very nicely. Enchiladas are a favorite use in our house for leftover meat from broth-making.

Here is a simple enchilada sauce you can make with ingredients from your pantry that is way better than canned sauce:

• Put 1/4 cup of paprika, 1 tablespoon cumin and 1 teaspoon ground
  cayenne in enough fat or oil to make a paste at the bottom of a
  smallish pot (about 3 tablespoons fat).
• It will be kind of stiff and crumbly. Us a whisk or spatula to move it around a bit.
• Cook it slowly on low heat until the spices begin to change color and give off fragrance.
• Add a jar of tomato puree or roasted garlic tomato sauce.
• Correct for salt and consistency, adding water to thin if needed.
• If the tomato sauce was chunky, use a stick blender to puree the sauce.

Mix half the sauce with your leftover meat. Use 1/4 of the remaining sauce to wet the bottom of a casserole dish. Fill or layer warmed corn tortillas with the meat/sauce mix and use the last of the sauce to top the enchiladas. sprinkle grated cheese if desired. Warm in a 350* oven until the enchiladas are fragrant and just starting to color on top.

I don't use meaty bones for my bone broth, but divvy up the pot this way: I get the broth, the dog gets any meat shreads, and the compost pile gets the veggies.

Reminder - don't give your dog cooked bones! Bones splinter once cooked - they only get raw.

Nothing goes to waste in this house. All meat, poultry, etc is saved once the bones are removed and is mixed in with dog kibble for a great treat and a healthy treat for our dogs. No added salt which is deadly for pets is a bonus.