Well, everyone had good points already, so I won't pretend my own comment will be any more valuable. But as often, I just like to spread my own two cents.
Pros :
-Quite obviously, this would allow people to use (and abuse?) their 9mm and knife way more often, as they are barely used except maybe in the first wave to save some ammo. Even the two perks which have access to skills upgrading their base loadout usually switch them after 2-3 waves MAX. Still, even if we consider it that way, it would obviously be a mobility tool more than anything... But I guess that's better than never bringing them out right? Plus, it would be even more valuable to the two perks which actually use the 9mm often (I'm thinking of the Zerk for those pesky crawlers and the Demo when he doesn't have access to something more reliable as a sidearm). Imagine a Zerk literally SPRINTING around the map. Damn.
-Out of pure nostalgia galore, this would be awesome. I guess at first, KF2 wanted to be more grounded in reality with his hyper-realistic weapon handling, near-future timeline and more accurate animation work. But then considering the surreal maps we got and the numerous wacky customization options, I believe moving faster with a knife wouldn't exactly be out of place. It makes little sense, sure. But not only is it a tacticool tool and trick to pull off, it surely would please the veterans.
Cons :
-As others pointed out above : speed has become waaay more meta than in the first game. Obviously, KF2 is faster-paced, and the zeds have multiple tools to prevent you from camping forever (this was the go-to strategy in KF1). This means that speed is way more valuable than it was in KF1 (even though kiting was also easier in that game : you only had to be careful of raging opponents and husks). Basically, sprinting, even with a fully-loaded character, is paramount. But at the same time : it makes the game way too easy at times. I'm not saying we should remove it entirely, precisely because it's so valuable. But it is true that against bosses or in open-maps : it can still make the game feel like a breeze when you know how and where to flee.
-Worse than that (but that was already true in KF1), speed is indeed a way to differentiate perks, just as much as skills or weapons ! And as I mentioned right above : speed is highly-valuable in KF2. So it's not merely a funky bonus : it is a real game-changer for the perks who have access to it. And again, I won't complain about it, as the ones who have access to it certainly deserve it. Let's take a quick look :
1) The Field Medic is meant to be the most supportive perk of the bunch. He also needs to heal... sometimes five people at the same time, people that may dispatch themselves in a fairly large radius ! As such, even though he got tracking darts, he definitely needs the mobility to always be on healing duty. It also allows this weaker perk to be more nimble and avoid the biggest chunk of damage ( "weaker" being a poor choice of words...He obviously became ridiculously powerful as he got new guns). But it also creates a problem with his extra healing topped with his speed , somewhat lightweight weapons and sprint capability : he can often kite forever. How many times have I seen a medic being the last man standing, circling around the map while damaging the boss... That's already a problem,so imagine if he had access to extra speed with his pistol or knife?
2) For similar reasons, the zerk truly need the speed too. He's always up-close-and-personal and thus needs to be able to kite reliably. It's frequent for him to get into the fray, soaks up a ton of damage for his team, only to fallback to heal a little bit. I think that's the one perk that deserves his speed boost THE MOST.
3) The gunslinger is meant to be a medium-ranged sharpshooter, and I think he does it very well. The consensus seems to think that he's one of the better perk right now, requiring very little tweaks... and I personally would agree ! And part of that appeal comes from his speed. But unlike the previous two who often use it to fallback, I would say the 'slinger use it more as a form of "combat dance", moving around the monsters, popping caps along the way. You often have to take risks as a GS, but if you master that so-called Combat Dance, you may leave without a scratch. It is, however, also helpful considering your guns have to be reloaded often. Sure, you can count that "one bullet equals one dead" with your deagles and .500, but let's not forget that you often have to empty your mags on a fleshpound or scrake to kill them. And similarly, being quick on your feet to dodge incoming attacks while you reload is great, considering you don't have anything else to protect you (the only resistance you got, "bullet", is highly situational, being helpful only against some bosses)
4) Finally, the SWAT with his lightweight weapons is also quite on the faster side. We had numerous talks about the perk in the past few months with some people praising him, but most of us lamenting his somewhat underwhelming gameplay. Just like the gunslinger, I believe it's part of his meta to be able to move around faster, downing dozens of trash zeds in a blink. Admittedly though, I believe his skill trees should be divided between "tankiness" and "speed", just as to be able to choose if you want to be a slower alternative to a zerk that has mad crowd control capabilities, or a crazy speedster that excels at hit and run.
All the other perks are slower, but it is part of balancing really. Could you imagine some "nuking" perks like the demo and sharp (and somewhat the support) with extra speed? You could pretty much destroy any incoming threat and worry very little about the repercussions. Usually, they struggle at short range, either due to poor magazines or lengthy reloads. Imagine crowd control perks like the Firebug or Commando with extra speed... Not only would them drop down nearly anything in no-time, but no zed could catch up with them !
Sure, to defend your point-of-view, you could say that you would NEED to grab on to your 9mm or knife to get that boost, meaning you obviously couldn't truly defend yourself with that. But my point still stand... Imagine you just fired a rocket. Part of the trade-off for such a damaging weapon is that you have to suffer a painfully long reload after each shot : reload which can be a death sentence if you don't pay attention (or if your teammates don't help you). But here, you could swap up to your 9mm, move further and reload safely. Rinse and repeat.
All in all, I don't think it would BREAK the game. As I said, zeds are more nimble and smarter than in KF1, so even if you flee : a husk could set your ass on fire or an EDAR Trapper could make you know your place. But I believe the fact you could kite way more easily (which is already a strategy that works a little too well) would be a bit too much.
... would be great as a mod though !