Can you have gluten free bread? I'd say you've got yourself worked into a corner here. Because of how gluten acts in dough, you either have to use gums to create that elasticity, or a blend of ingredients that can mimic some of the properties.
Gluten free baking, particularly yeast bread, is a completely different proposition than its gluten-y counterpart.
If you don't want to use "a million and one different flours," then a pre-mixed flour blend is your best bet.
However, most pre-mixed flour blends contain gums. If you don't want gums, you should expect that you're going to have to use multiple flours.
Personally speaking, I make my own blend, based off of Shana Ahern's recipe. It's six different flours: brown rice, sweet rice, potato starch, tapioca starch, almond flour, and sorghum (or teff, or millet, or buckwheat) in a 40% whole grain/60% starch ratio. I make about 5lbs at once, for significantly less than the cost per ounce of the ridiculously overpriced Bisquick/Cup4Cup/King Arthur/pick-your-brand-since-they're-all-the-same-three-ingredients-anyway. Tastes a whole lot better, too. Is it a pain in the ass? Yeah. It's better than paying $5 for less than a pound of flour, though.
I also like Shauna's recipe for multigrain yeast bread. It's a good hearty bread without being gummy or otherwise funkily textured. The flaxseed/chiaseed slurry makes a much better substitute for gluten than any gum I've ever tried, and it adds a nice nutty taste to baked goods.
Most pre-made gluten free flour mixes are a mix of brown rice, white rice, tapioca starch and potato starch. It may be difficult to find pre-mixed GF flours without xantham gum premixed with it, so you may need to mix your own flour. Here's a good, simple, starting mix. You can add in other grains later for flavoring as desired. Millet, Buckwheat, Amoranth, Sorghum, are all good choices to experiment with.
Some Tips: Finding the right amount to add of the binding agent you choose may be difficult, expect to experiment, but always start with the recommended measure on the packaging. I have no tips for anything other than Xantham Gum, which is to not use too much – it gets slimy and rubbery. Too little, and your dough will crack when rolling or kneading.
If you do yeast/egg breads, use an extra egg – I don't know why, but it yields better results.
Give the yeast longer to rise than wheat.