I am amazed, my American friends won't eat innards as they call them. Of course we call them offal. Liver is lovely if mixed with minced (ground), pork and made into pate!
We have probably the same cuts of meat as you, but with different names. Shin of beef is ideal for pie making, when cooked slowly first, then added to the pastry. We have stuffed hearts and of course tongue that is pressed and eaten cold in sandwiches, or on a meat platter.
Many pubs have Ox roasts on a Sunday. Ox is a breed of cattle and quite delicious. They spit roast it and you get an awful lot for your money. It also makes Oxtail soup, a classic soup, and the taste is incredible.
My Grandma told me that the more work a cut of meat does on an animal the better it tastes. Although it also takes a lot longer to cook,it but is worth it.
Rib of beef makes the most fantastic Sunday dinner, with Yorkshire pudding. The gravy and beef dripping is to die for.
Saltmarsh lamb is very expensive and the ultimate. For the sheep do not eat grass, just what appears on the seashore like seaweed etc. The taste is something one must try.
Regards.
Neil.