I grew up hunting with a group of cardiologists. It was fun cleaning deer with them. We discected and learned about every part inside their chest cavities. (Tried to leave the guts alone). One particular heart surgeon tried to not let anything go to waste. He’d always make head/neck shots so as to keep the heart, liver, and even the kidneys. He was also an excellent Italian chef. I don’t ever remember him trimming off the heart membrane; he just got it really clean, and prepared same as mentioned above.
I’m not a big fan of liver, but man, that Doc could make deer liver and onions that anyone could enjoy. Seriously, deer lease meals were always fantastic.