Without wishing to knock Glenkinchie, surely for distilleries you are looking at the Highlands or Islands, not Edinburgh?
The last few times I have been to Edinburgh I have watched some rugby at Murrayfield. Probably more a case of the reason why I have been going to Edinburgh is to watch some rugby.
Take a bus tour, walk the Royal Mile, and see the Castle. Edinburgh is one of Europe's top cities with great architecture and wonderful views. OK, the locals have a bad reputation for snobbishness, but, much like London, the number of visitors and immigrants (i.e. non-locals) ensures that you don't have to meet too many.
I'd add visiting the Royal Yacht Britannia at Leith and having a trip round the Scottish Parliament as too further options.