Gatwick is far more accessible by public transport than Heathrow. Sure road links would need improving but that goes for any airport expansion. But in terms of public transport Gatwick has much better infrastructure than Heathrow. You can only go to 6 places by Heathrow on a train compared to 150+ from Gatwick – which will increase in 2018 with Thameslink upgrade.
Also far more people access LGW by Public Transport than LHR by a significant amount in % terms of those using the airport – so it is Heathrow at the moment which is very weak in this area.