I use my overdrive pedals (one at any given time on the board) as "spice" for a song here or there - most of the time I play without them. I play church gigs, and it's rare that you want dirt for every song in a set (it has happened, but it's been a couple years since that happened). Anyway, they are still important - if you need some edge/extra energy for a given song (often the opener), you need some dirt.
I have a couple Darkglass pedals (B7K, Vintage deluxe), and a YYZ. All of these can be set to as subtle as you want, and they all have clean low end. The Darkglass pedals have 3 different filters you can use going into the distortion circuit, and 3 coming out - you have 9 different overdrive flavors in one pedal the choose from, and a 4 band eq on the combined signal (distortion plus clean). Despite all their demos sounding the same and very aggressive, the reality is that their pedals can do a LOT of different sounds. The sound in my head is VERY far from anything you hear on a Darkglass demo, yet it's fairly easy to dial "my sound" in - you have to figure out a couple things (notably, as you dial in more distorted sound, you lose some low end, but the eq on the sum signal works great for restoring that), but after that, it's pretty straightforward.
I sound like a Darkglass devotee. Well, I am now, but it took me a while to get there. For years, I bought cheaper overdrive pedals, returning some and keeping others (online demos don't tell you what it will sound like in your rig), but never wanting to spend the amount of money that a Darkglass pedal costs. I ended up with a shelf full of overdrive pedals, and a few blending pedals to make up for the deficiencies of the pedals I had. Last year, I sold off my cheap overdrive pedals, and (surprise!) I had more than enough credit from that to buy the 2 Darkglass pedals, and the YYZ. If I had started with the Darkglass pedals, I would have saved myself a lot of time and money - they are flexible enough that, as your tastes change, you can dial in another sound, and not have to buy even more pedals.