There's additional issues with changing channels than just GOP sizes. For cable TV signals are still channelized into 6MHz channels. QAM modulation is used which gets about 40Mbps per 6MHz channel.
An MPEG transport stream is sent over that 40Mbps "channel". Inside of that transport stream are several elementary streams, each carrying either video, audio, or data/program information. A TV station will be carried by a combination of a video (MPEG-2/4) and audio stream. These streams are packetized such that a receiver won't get buffer underruns or overruns. The transport stream is further packetized. Adjacent station broadcasts aren't always carried on the same transport stream.
So when you hit the button to "change channels" the cable box may need to tune to a new radio channel, wait for transport stream packets for the station, and then buffer enough data to start playback.
The GOP size then plays a part because the decoder needs an I-frame to do anything useful with the P-frames inside the GOP. Cable providers use longer GOPs to get better quality at a given bitrate, I-frames are much larger than P-frames so the more you send the higher the bitrate you need.
Sending I-frames for other broadcasts in the same transport stream would be a waste of bandwidth. The data used for these rarely used extra I-frames takes away from bandwidth programming in the stream could use. There's also still the lag from tuning a new channel and waiting for those I-frames to show up. So you've wasted bandwidth for a very slight increase in responsiveness.