Stove burners too hot to simmer

Even if you solved the sauce problem by braising in oven, you may still want to control stovetop heat.
There is such a thing as a heat diffuser, to put between the flame and your pot.
Look it up on Amazon usa or uk so you get an image, then look to buy it near you.

This is going to sound a little off the wall, but when I lived in Italy, I had the same problem. In desperation, I devised this solution.

If you have access to heavy duty tin foil (I had to bring it back from the US), ball up four handfuls. Make a four square away from direct flame but so the pan can rest on them securely.

It should put enough distance from the flame to lower the heat.

Be careful that they’ve cooled adequately before you pick them up or dispose of them.

Lori T has it mostly covered, but here's another idea:
Depending on your hob/grate shape and what else you have on the stovetop, is it possible to slide your pot over so only part of it is above the direct heat of the burner? I do this often on higher powered stoves and it works fine to reduce the absorbed heat.

You can certainly do a braise in the oven, I have. I use a 300F oven, and leave the lid off the last half hour or so, to allow it to thicken some. There really isn't a problem with the meat overcooking, since the cuts they use are really well suited to braising using whatever method works best for you. You could continue to do it on the stovetop, if you use a cast iron skillet or griddle over the flame itself, and sit your pot inside or on the cast iron surface. On your lowest flame, it will help tame and distribute the heat more evenly and gently. Finally, you could opt to use a slow cooker- although you would want to start with a little less liquid, since it will not be evaporating at all. This method also takes the longest- about 8-10 hours on low. I personally like using the oven, and an enamel/cast iron pot with a lid to do braises of all sorts, including the one you are attempting. Braises in the oven are nice because the temperature is constant, all around the pot, and you don't need to pay much attention to it while it cooks. Oven braising is much easier and gentler than trying to regulate a stove top flame, or even an electric element. So you can pick whichever suits your needs or equipment best. Good luck, and good eating which ever you go with!