There really is very little point is just building a 3rd runway at LHR and then filling it with flights.
Part of the issue with LHR is that it is running at 98% of capacity compared to 75-80% at FRA, AMS and CDG which gives very little flexibility when there are any sort of problems. It might also improve the on-time records of airlines too (though of course not all on-time issues are down to the airport most are with the airlines and their passengers)
There probably needs to be another terminal as well - perhaps a mega one serving US flights only that could (obviously with Governmental approval) offer pre-clearance
So yes use some of the new capacity for slots for new routes - and they must be for new routes only and especially to places line India, China and South America and if an airline can't make that route work then they have to give them up and not just swap them to another one. That might spur some actual competition between existing and new airlines.