Using different vehicles wouldn’t have made any cost savings - the coach exists, and whether it is used or not, the horses that pull it have to be fed, exercised and cared for. And one could argue that horse drawn vehicles are greener than a fleet of limousines.
When there are State visits, transportation has to be provided, so why not use resources that the country has already? You can’t expect Heads of State to arrive at Buckingham Palace by bus. And doesn’t it impress visiting dignitaries, when the red carpet is rolled out for them?