Josh, both cooking in the oven and on the stovetop work perfectly. Le Creuset and Staub etc. don't make "special kinds" of oven pans like you mention. Those pans are made of cast iron (melted iron is poured in a sand mold to make them) and are ment to be used either on the stovetop as in the oven. Most of our grandmothers or generations before that, used cast-iron pots and pans to cook with; there was simply nothing else and they lasted a lifetime and longer.
Nowadays we use more modern inox aka stainless steel pots and pans; they are substantially lighter than cast iron. But, inox mostly has handles that are made of materials that cannot go in the oven! That's another important difference with cast iron cookware. Also, the thick cast iron distributes the heat more evenly over the whole surface of the cast iron pots, bottom, walls and handles. It takes a lot more time to heat them up but once they are going, they will buffer short bursts of heat or cold.
My own grandmothers (I'm 65) both used cast iron on stoves that burned coal. They mastered their stovetops perfectly by pushing their pots backward or nearer towards the heat source. Making stews like boeuf bourguignon or the most popular stew around here, carbonnade flamande, were all done on the stovetop on low heat and in a cast iron pot.
You will probably find the same amount of people swearing by cooking stews in the oven as there will be people cooking stews on the stovetop, both have advantages. I always cook on the stovetop, sometimes in cast-iron, sometimes in stainless steel. The advantage of cooking on the stovetop is that you can watch what's going on permanently by simply lifting the lid. Opening the oven causes a lot of heat to escape and checking what's going on in a closed pot in the oven is quite difficult.
When cooking on the stovetop, you can also easily taste and check for doneness. You can play with the lid which is important in making stews; towards the end of the cooking time you might have still too much liquid, so you let the lid off or leave a little opening, to reduce the liquid slowly.
BTW, you don't have to go for a Le Creuset or a Staub when you want a cast-iron pot. Look for the no-name pots; most of them are made in China or so and are equally good! They cost around 1/3th of a Le Creuset or Staub.
Have fun, boeuf bourguignon is a dish that should be on every amateur cooks "carnet de cuisine".