Symbolic perhaps, but useful, no. Does anyone fly to "John Lennon Airport" or "George Best Airport"? No they fly to Liverpool and Belfast City. To rebadge LHR would simply result in more corporate expense for very little gain. Would anyone choose "Queen Elizabeth II Airport" over Gatwick because of some kind of allegiance to our former Monarch, albeit that she was globally known and respected? No, they'd fly to the most convenient and / or cheapest from their perspective.
When commercial aviation was novel airports got named after famous people such as JFK, but these days people are more interested in where they fly to, not what what the place is named, that's why places like Elmdon, Woolsington, Yeadon, Ringway etc have been consigned to the memories of old farts like us.