The Ffestiniog used to run a half hourly service for much of the summer, and does still run some very intense services on special occasions.
As you say, the GC should in theory be able to operate a very frequent service over its double track. However, in practice I suspect that they are limited by how many trains they can despatch in quick succession from Loughborough, and how quickly they can deal with them at Leicester North.
The Manx Electric, another double track line, used to run a very frequent service (as I recall, the timetable specified a minimum of half hourly with additional trams as required) as they could despatch trams in short order from Douglas Castle, and turn them around very speedily at Ramsay, but sadly those days are long gone. The Volks Electric claims to run a train every 15 minutes.
In general, the shorter the journey time, the more journeys a train will be able to run in a day. We saw this with the Welsh Highland, as the line got longer, so the service got less frequent. A line with a very short trip should be able to pack in a lot of journeys in a day - how about some of the miniature railways, such as the one that runs in the park in Chester? I suspect that they must be able to pack in a lot of journeys in a day.
However, the OP asked for most intensive, not most frequent, service. Not sure how you measure intensivity, but Harbour Station on the Ffestiniog is quite intensively worked with regularly 4 snd sometimes 5 trains in the station at the same time. If you are talking about the whole line, I suspect that the Volks is going to be a top contender again.