For sake of not arguing, I won't ask why anyone would categorically overcook all proteins. But if you do, you have cooked out almost all fat & moisture. It will now have nearly the fat content of totally lean meat. It's the fat that give it moisture, taste & mouthfeel. Cook it all out, and you just have dry strands of protein.
Weigh the meat before cooking, then again after cooking. The difference is fat & moisture.
If you want to identify just the fat, try this. Weigh your patty then put it on a rack over a small pan. Cook it, either in an oven or on a grill, until well done. The rack will allow the fat to drain, and the pan will catch it. A lot of the moisture will evaporate, but at those temperatures, the fat will only melt, not evaporate. Now weigh the dry meat patty. Then chill the drippings from the pan so you can easily separate the fat from the water. Weigh the fat. You will now know how much fat actually cooked out. This won't separate saturated from unsaturated, but most beef fat is saturated. Now, if you add the weight of the cooked meat to the weight of the fat, and subtract it from your original weight, you'll also see how much moisture you lost, and why your well-done beef patty is so dry. (Sorry, that Harvard food science class is on my brain).
This all begs the question, Why are you so worried about the saturated fat levels? I assume (I know...) that you are concerned about diet & cholesterol. Recent studies have debunked the idea that saturated fats are necessarily the evil that they were purported to be in the 1980's. Rather, it's trans fats that are proving more harmful. There are also indications that genetics are proving to be a more critical factor than diet. Genetic disposition doesn't respond well to dietary adjustments, but does respond very well to medication.
Here's some recent reading on the subject. Granted,
some of these sources are more reliable than others, but there's a lot of new information available about diet & cholesterol. Read stuff published since 2013 & 2014, not the old stuff.
Don't Fear The Fat: Experts Question Saturated Fat Guidelines : The Salt : NPR
Saturated Fats: Should I Eat Them or Avoid Them?
New Science Destroys the Saturated Fat Myth
How did we come to believe saturated fat and cholesterol are bad for us? - The Eating Academy | Peter Attia, M.D. The Eating Academy | Peter Attia, M.D.
http://well.blogs.nytimes.com/2014/03/17/study-questions-fat-and-heart-disease-link/
Time Magazine: We Were Wrong About Saturated Fats
Saturated fat's role in heart disease is a myth, says heart specialist - Medical News Today
Saturated fat is not the major issue | The BMJ
For more personal anecdotal information, ask several folks here what happened to their LDL when they went on high fat, low-carb Keto diets for diabetes & weight loss.