Why isn't this VCO responding to my sawtooth waveform?
Your WiFi radio is essentially a packet radio, and transmits and receives packets (called "frames" in the 802.11 protocol) of data when there is data traffic. 

When you drive the VCO with a sawtooth, if your linearly-sweeping VCO output happens to hit your WiFi channel frequency band during a ping packet transmission, it may cause an error in that packet, and the WiFi protocol will then request a retransmission, by which time the VCO frequency will have "moved on" to a different WiFi channel. Because the VCO sweeps relatively slowly (compared to the ping packet transmission rate), your chances of hitting a packet are quite low, unless you are almost filling the channel with packets (like downloading a large file from your local network drive).

However, when you drive the VCO with noise, it's frequency hops around really rapidly, and the chances of it hitting your WiFi channel frequency during a ping packet transmission are greatly increased, hence the consequent drop of your WiFi channel.