If he thinks that's an argument against LTNs, then he's mistaken – it is actually a clear example of how LTNs encourage people out of their cars. As well as keeping through traffic out of local side streets, LTNs also make short car journeys less convenient. This encourage drivers to reassess these journeys, and make better decisions.
If a journey is now, with the LTN in place, more convenient when done on foot or on bike then more people will, over time, make that choice and, crucially, fewer journeys will be made by car. The traveller will be fitter, the air will be cleaner, the roads will be safer (for people on foot, bike, and mobility scooters) and there will be fewer traffic jams – everyone wins. Very importantly, the few people who really do have to make even short journeys by car will not be held up by all the unnecessary cars that currently clog our roads.