Good answers above, so I won't repeat those points; I largely agree
I would also point out that it's not necessary to just drown your model when you apply washes. For most of my models, it's quite the opposite: I paint in my washes where I want them, the same way you would paint details if you were painting in a darker color. I use a pretty fine brush (size 0 or smaller), and control how much/where it goes. The reason I use a wash instead of a darker paint is because washes are more transparent, staining the original color but leaving more of the original color than a thinned paint. Because of their nature, when they dry, they leave a more diffused edge, instead of a hard one, without a lot of work. If I used a dark paint, I'd have to feather the edges or wet-blend them in, which is a lot of work.
Incidentally, when you use a triad system like Reaper, you don't HAVE to start with the middle color in the sequence. You can also start with the darkest color and work up (or if you want, the lightest color and work down...).
Generally, there are practical advantages to starting with the darkest colors. Most painters will agree that miniatures are generally easier to paint by finishing the hardest-to-reach, deepest areas first, working up to the easiest-to-reach, most exposed areas. It's just a physical limitation; if you paint the stuff on the outside first, and then try to reach the stuff tucked away, you'll get paint on the parts you've already painted. Although there are plenty of examples of good work to the contrary, the reality of most gaming miniatures is such that the recessed areas are darker, and the outer, raised edges are most highlighted, because that's the way our brain thinks that light would naturally hit an object.
So, following that logic, in the broadest generalization, it's often easier to start with the darkest paint in the deepest areas, working up to lighter paint in broader areas, and then finishing with the lightest highlights on the raised edges. When you do the
GW recommended method "base + wash + layer + highlight", that's essentially what you're doing.