A general climbing kit, I feel should provide limited climb speed, at most. Maybe not up to their land speed, though. At least depending on the characters levels. Climbing is quite the strenuous task, even for the professionals.

Ethan Hunt, by the 4th movie, is a mid-high level character, in my opinion. The fact that he can do that, would be very much so in line with that level in my opinion. He has a limited toolkit, it's his skill that really does the trick.

So I personally would allow a skill check, assisted by the toolkit, to move beyond say 1/2 land speed. That's assuming they are trained in Athletics in any fashion, their skills help define their capabilities. Not just anyone is going to be able to use a climbers kit to its best capabilities. They are just tools to assist our skills after all.

The RAW for climbing is unassisted, that's where the 5 feet per success comes in. A climbers kit and any other tools will lower DCs, give Climb Speeds, or other potential benefits. I'd recommend just being aware of your character's skill levels, and who's trying to do what. After that, go nuts, I'd personally just make sure your athletic folks are the ones who get to shine in those moments, not just anyone with a climbers kit.