Run Speed Up does not affect any of the Roller's speeds while running with them out.
Ah. Oh well, it was a nice thing to imagine.
After thinking on it some more (and reading some other people's thoughts), this thing is probably pretty bad for combat. Even if you snuck up on someone, I feel like this thing takes long enough to splat enemies that anything short of a Dynamo Roller or another Inkbrush could probably turn around and splat you, if not with their main weapon, then with their sub weapon (or the Inkzooka, or a non-Suction-Bomb Bomb Rush), or they could pop a Bubbler or Kraken and at the very least ink a path out of the fight, or at most splat you. The Sprinkler-and-Inkstrike loadout doesn't help here either. Maybe it's just personal preference, but I would prefer a different loadout, perhaps something like Burst Bomb-Inkzooka/Bomb Rush/Bubbler (Burst Bombs could be useful for damaging enemies to put them in range to splat them within 2 or 3 swings, and they don't cost much ink; Inkzooka would give you a way to fight enemies who are shooting you from outside your seing range, Bomb Rush would just let you get the benefits of Burst Bombs with sort-of-rapid-fire, which seems like it would be pretty neat, and Bubbler would let you survive fights if things don't go as planned or allow you to attack enemies with high-power weapons like rollers or .52/.96 Gals).
But I'm not sure about its potential for inking turf, either. True, it goes fast, and its swing has pretty wide range, and Sprinkler lets you ink an area and then go ink somewhere else (do wish it didn't go away when you died though), and Inkstrike lets you just claim an area whenever you feel like it, but I don't like how weak it is. What's the point of inking all that turf if you'll have to run away, hide or fight till you get splatted (and in my experience, you WILL get splatted if you fight with the brush) the second anyone with a roller, or a .52/.96 Gal shows up. Also, it eats through your ink really quickly, and Sprinkler doesn't help much with how much ink it eats per toss (I'm just glad it lasts forever unless you get splatted or toss another one).
Maybe if you can rely on your teammates to keep areas while you go around inking their stuff, or something like that, I could see this being a quality support/harassment/turf-claiming (is there a word for that?) weapon. Though then if they get overrun, you're probably not capable of assisting them very well in reclaiming turf (like I was saying earlier, someone with a weak weapon with short range seems like they're gonna get splatted when they try to attack).
For now, I'll hope for an Inkbrush Custom/Deco with a different loadout (hopefully, it's a bit more combat-capable, or at least has Bubbler).
But that's just my ponderings on the matter. I could be wrong.
Reading this over, I think I'm thinking mostly of Splat Zones right now; not entirely sure about its Turf War usefulness, but I think most of what I've said here applies there too.