Putting the "not at weekends" matter aside for a moment - I have to say that I think I would always choose to go from Yorkshire to Gatwick by train. Whether you come into Kings X or St Pancras, it's easy and convenient to get from there to Gatwick on the FCC service from St Pancras.
As for weekends.....the cross London direct FCC service is suspended on MOST but not all weekends. If it is not running on your date, there is an alternative route by rail. I don't think anyone else mentioned this - forgive me if I missed it. It has a very easy connection, not in London, but at Reading. In the regular service pattern, direct trains run from there to Sheffield, Doncaster, York and points north. And there are direct trains from Gatwick to Reading. So one change will get you to Yorkshire, though not actually to Leeds for which another change will be required at one of the above.