The most obvious thing is it keeps your oven free; a range has only one oven, but four or so burners. Its also often easier to check on it, add ingredients, etc. stove-stop. Stove-top also lets you quickly turn up or down the heat. Stirring is easy stove-top, more annoying in the oven.
For what you're doing, the biggest difference is going to be where the heat comes from. Stovetop, of course, comes almost entirely from the bottom of the pot, the lid for example can remain fairly cool, meaning moisture evaporating can condense on it, and drip back in. In the oven, the heat surrounds the pot, so the lid will be just as hot—or hotter, even—than the base of the pot.
To take a recipe designed for stovetop, you'd perform all the browning steps stovetop (of course). Maybe add a little more liquid. Then bring it to a simmer stovetop, and finally transfer to the oven.
Putting a piece of aluminum foil over the pot (under the lid) can help keep moisture in.
In the food safety section of Modernist Cuisine (1-138) it says that all wild rabbit must be cooked through:
Wild rabbit must be cooked well done to kill any tularemia bacteria; for rare or medium cooked rabbit dishes, chefs can substitute farm-raised rabbits that have been kept segregated from their wild relatives by trustworthy breeders.
The section on meat (3-96) recommends bringing rabbit loin to an internal temperature of 59C (138F) and going by the section I've just quoted this is obviously only for farmed meat. The recommendation for the legs is to cook sous vide for 1 hour @ 66C (151F) which again obviously refers to farmed because that's definitely not long enough to tenderise wild rabbit meat.
I would really recommend cooking the leg meat and loin separately as I find that the loin dries out long before the legs are cooked through. The legs are much nicer braised too.
Best Answer
As far as I know, Collagen starts to break down below 60°C/140F, time play a big role, i.e. to get the same breakdown at 60°C as 80°C you might need 24-48h instead of 3-4h.
The higher the heat the more the meat is contracted and it will get dryer, in a stew that might not be as obvious as in other cooking methods but it should still be the fact. Compare tough cuts of meat cooked sous vide at 55°C-60°C for 24-100h, they get very tender and moist.
So the benefit will be moister meat. I think you must cook the veggies at much higher temperature (80°C-95°C) so you would have to separate the two. If the end result is worth the effort I do not know, I have wanted to try doing a 'stew' with meat cooked sous vide at 55°C - 60°C for 24h+, with the veggies done on the side at 80°C-95°C, normal stew style, to see if it would be better, but I have never tried.
Sources
Hum, I'm sure you or anyone can find enough sources to argue that you need 70-80°C(160-180°F), see e.g. On food and cooking(2004) pg 163 "...Meats with a significant amount of tough connective tissue must be cooked to a minimum of 160-180°F/70-80°C to dissolve their collagen into gelatin. "
The only potentially credible and official (what do you mean by that?) source that potentially can trump On Food and Cooking is Modernist Cuisine, I will quote.
MC 3.78-79 "When meat is cooked at temperatures above 50°C/122°F, a different chemical reaction -- the conversion of collagen into gelatin -- becomes prominent." it continues and says "Cookbooks and some scientific sources commonly claim that collagen converts to gelatine "at" some particular temperature. Usually the cited threshold lies between 60°C and 75°C / 140°F and 167°F. In reality, the hydrolysis of collagen is simply another example of a chemical reaction whose rate varies exponentially with temperature." it goes on describing that the exact temperature have not been measured and that it depends on what type of collagen, e.g. breed, cut and age. Finally it says "Tough meats can be cooked until tender at any temperature above 55°C/130°F (and even a bit below, if you are patient enough)."
Also see lots of sous vide recipes on the net saying that you can cook tough meat at much less temperature than 70°C/160°F and get it very tender. Or try it, I have and it works!!! I have stopped eating primary expensive cuts, flank, shin and brisket tastes much better!
See also SAJ14SAJ reply and Stefanos comment below, as per Baldwin which should be almost as good a source as oFaC and MC
Conclusion
The answer to the question, "how low can you go and still get collagen break-down" I would answer 55°C/130°F or even a bit lower. That is NOT a realistic temperature for cooking a stew, especially not if you will put veggies in it, but as far as collagen breakdown goes that is the lowest temperature for 'stewing meat'.