A small museum and library would be located within the walls of Burthorpe. Because it is capital Burthorpe, the RS gold school could be located nearby.
Another problem that is a problem with Burthorpe is that it's the only bank that is located in Burthorpe (I'm not including the Warriors Guild as part of Burthorpe) which is in the Thieves den, not good for a capital city, eh? There would be North and South banks.
Is there any city attraction? How about a Minigame at the Barracks. You could also have a good smithing OSRS buy gold center for creating weapons and armour. Good specialist shops would also be great. Let your own choices be made. We appreciate your attention!