I understand what you're saying -- this is about the right answer, not who's right
. . . and I learn something new every day . . .
So from looking deeper into the protocol, the answer is yes and no.
Most devices do actively scan for a broadcasting client once per channel (I didn't know that), ironically, in the neighborhood of 25 to 50hz (a bit slower than most of our radio's update rate). While they can scan passively, that mode is about 4 times slower, so the devices will kick on their transmitter in active mode for short bursts when you ask to find hotspots in the area -- it's not quite evident if they stay "active" if you aren't seeking these connections, and the answer appears to depend on how the device is programmed.
So Yes, there will likely be more traffic with a room full of smartphones with Wifi turned on but disconnected than if the wifi is turned off. By how much would depend on how the devices are programmed.
. . . but . . .
Wifi in general practices CSMA/CA (carrier sense multiple access with collision avoidance) -- a big mouthful to represent "how I keep from stepping on other wifi radios I can't hear, but are in range of a router". Effectively, before any wifi radio transmits, it listens. If it can't hear anything, it then sends a brief burst . . . and stops. How brief is this burst? depends on the bitrate -- the probe request is 2 bytes long and at a 2Mbps rate (which is a typical slow rate) that's roughly 1 microsecond . . . every 20-40 milisecond, and only if it doesn't hear something first. That's a duty cycle of 0.005% to 0.0025%. Larger packets (like big chunks of juicy data) take much longer, but if you haven't connected and there's nothing out there to connect to, your packets don't get any bigger.
Long story VERY short . . . it would take MANY (thousands) cellphones packed within a small area to deny access to 2.4GHz to a non-cooperative RC transmitter/receiver pair. Now if there's a high capacity public wifi hotspot available within range . . . all bets are off.
How many control transmitters would it take? Regardless of the frequency hopping scheme, I doubt they can share the band between thousands of them . . . but you'd need to know the bitrates of each to be accurate.