Of course we drive up!
I'll slightly modify the sentence: "Of course, we drive up as close as self-protection allows." Apart from that, in AGT, there is no sprinting; it's hardly possible as the first attack team with all that gear.
A movement area for the fire brigade is usually not directly at the house wall either. One would have to look at the local situation. I know the overzealous platoon leaders who drive up right in front of the house so that afterwards no one can get forward or back anymore ;)