A library and small museum would be located inside the walls of Burthorpe. Because Burthorpe is a capital city Burthorpe, the RS gold school could be nearby.
Burthorpe is also plagued by a problem. There is only one bank that that is located in Burthorpe, I'm not putting Warriors Guild as part Burthorpe. It is situated within the Thieves den. This is not good for a capital city. There are North and South banks.
What city attractions could you imagine? What about a Minigame in the Barracks. A good area for smithing to make weapons and buy RuneScape gold armour. Some specialist shops would be an excellent idea as well. Let your own choices be made. Thank you for reading!