If you are cooking beef fillet, NY strip steak, or Porterhouse steak, then there are several approaches that don't need a thermometer. One of the recommended approaches is to press on the meat with one finger and see how it resists. And you had some more suggestions. And you can cook such meat over a very hot fire that browns the outside and still have the inside not over cooked. With such tender meat, can forget about melting out the collagen since there is so little of it.

E.g., I have a Beef Stroganoff recipe where use beef fillet, cut the meat across the grain as matchsticks, and fry, really just toss quickly, until no longer red and DONE. No thermometer needed.

And can do some similar things with chicken breast, pork tenderloin, etc.

But for tough cuts of meat, you must melt out the collagen. With the meat at 160 F, that can take hours.

With a thermometer in the meat and frequent and careful adjustment of the heat source, you can keep the temperature near enough to 160 F to be successful.

Without a thermometer, you are in deep trouble: If the meat temperature for all those hours is too much under 160 F, then you can have a food safety problem. If the temperature is too much over 160 F for very long, then you can overheat the proteins in the muscle fibers and totally ruin the meat. That's what I did, dozens of times, all disasters, until I learned about 160 F. I'm still simmering angry over the totally sick-o information I got from the NYT, Julia Child, Jacques Pepin, etc. that caused me to waste all that time, money, effort, food, etc. A LOT of waste. Bummer. I was boiling oil angry.

In practice, and the hope of recipe publishers and authors such as the NYT, Julia Child, Jacques Pepin, is to make a stab and hope: Here is how that goes. Assume the cook has a fairly standard stew pot (although I'm not sure there is such a thing since there can be stainless steel, cast iron, and cast iron coated with enamel or Teflon, etc.). Assume that the meat they are stewing is beef chuck -- that is, with relatively little collagen. That is, no way will the cook be using flank steak, eye of round, bottom round, or even top round. Assume how much volume the cook will have in the pot. Assume that the pot has a good cover. Assume that the ingredients are at room temperature when they go into the oven. Specify that the oven temperature will be 225 F, and assume that the oven temperature control is accurate. Specify the cooking time. Hope.

Then with all those assumptions, there is a shot at getting good results. Why? Really, just by happenstance and good luck. Because as the temperature rises because of the 225 F oven, there will be about the right time at 160 F or so to melt out enough of the collagen (beef chuck roast doesn't have much collagen) and, as the temperature rises on the way to the oven temperature or to boiling at 212 F, the temperature actually won't get much above 180 F or so for very long so won't ruin the proteins.

Some of the recipes will specify a "gentle simmer" -- that's just psychological hog wash as a security blanket and a fraud and an invitation to disaster: A simmer is close to 212 F, and that's is WAY too darned high -- beef anywhere near 212 F for very long will be RUINED.

The only temperature being considered is that of the oven on the oven temperature setting, and that's darned poor control.

Change any of these assumptions, and will likely end up with disaster. I did dozens of times.

So, really DO have to use chuck roast. Actually, at one time, I called the HQ of a beef industry group -- they said to stew chuck roast.

With that NYT, Child, Pepin etc. happenstance based fraud, the common idea that are using stewing to make tough cuts of meat tender is set aside; if actually use fraud on a tough cut of meat, then the meat will be too tough at the end time of the recipe, and if keep cooking the meat at the oven temperature of 225 F the meat will get too hot and the proteins will be overheated and the meat ruined. I did just that dozens of times.

As in the OP, sure, the main secret of cooking tough beef is "low and slow", but actually to do this without being too low for food safety and without being too high for the proteins, really MUST have quite accurate temperature control. Thus, "low and slow" is just easy to take, light entertainment, at best worthless and otherwise misleading.

Sure, a good cook with a lot of experience and just a wood burning stove or just an open fire and using a lot of surrogate measures of temperature could be successful. But a thermometer and the crucial number 160 F is much, Much, MUCH to be preferred, really, now are essential.

But the VEFEEE authors and publishers very much do NOT want to bother you with 160 F so just pass out their fantasy nonsense that, however, is, really a fraud. I'm still simmering angry over the fraud.

Net, for beef stew of tough cuts of meat, a thermometer to measure the temperature of the meat is just crucial and, really, a great solution to a nasty problem. We should welcome the solution, not resist it.

And for much of BBQ, the situation is the same.