How to to prepare red deer haunch without using high temperature, so that its not dry? I am trying to perfect the method of preparing red deer haunch, and so far I was unsuccessful to make a really good one. I already learned to tenderize it with vinegar/papaya/kiwi for 24 hours before cooking, and although it makes it more tender, after cooking the meat still comes out dry.

What method of cooking to use to so that it doesn't come out so dry?

Because of health concerns of advanced glycation end products (AGEs) I'd prefer to use as low temperatures as reasonably possible while still achieving a tasty meal. Similarly out of health concerns, if possible I'd also prefer not to use alcohols or sugars during preparation.

Some tip I've been told is to completely immerse the meat in goose fat and cook on low heat several hours. Is this a good idea or is there a better way to do it?
As far as i'm concerned the secret is low temperature and patience. First of all, let's state that when cook a piece of meat as lean as red deer haunch, it is truly hard work to avoid a dry texture, given how little fat there is in your base product.

The way I handle this type of cut is to quickly sear it and then put it in an oven below boiling temperature for a LONG time. Just keep it at about 90 C and it will be good after about 5 hours (and still be good after 7)

Make sure the meat is covered in stewing liquid and there should not be any loss of moisture from evaporation. Give it that much time and the meat will become tender and moist despite missing the fat.

What I do to avoid watery flavour is to spoon some of the stewing liquid (about a liter) out into a separate pan an hour before serving and reducing it all the way down. You can then work it up to a sauce to have a creamy texture tasting like a concentrated versionof the stewing liquid. This will work regardless of the what you put in the stewing liquid, and does not require alcohol or sugar to produce a flavorsome sauce.