No way would I fly part way around the World to Heathrow to collect my baggage, get on a train, check in at Gatwick, go through security to get on another aircraft to fly to my final destination.
A fast link that allowed terminal to terminal connections between LGW and LHR could actually work quite well for transit passengers. If it was an airside transit link then it would work for passengers from domestic flights transiting to international flights.
A 30 minute frequent airside link wouldn't be so much worse than the inter-terminal transfers at LHR over the years where a 150 or 180 minute connection time could easily be consumed by the queues and congestion.