I'd have thought that EDI could support a similar long-haul route network to Dublin given its status as a major financial centre and a popular tourist destination.
There are a number of reasons why Dublin has a superior longhaul network. Most importantly, Dublin has a much higher population than Edinburgh, it has roughly 3.5 times the population. Its also much more ethnically diverse (people who fly home). Also while Edinburgh is a financial centre, Dublin has far more US companies based there, as Ireland is a corporate tax haven and lots of US companies are based there. Additionally lots of Irish people have emigrated to the States over the years and they like to visit. And Dublin acts as a central airport for the whole island of Ireland with airlinks to most other Irish airports in a similiar way to Heathrow. Dublin also has less competition than Edinburgh. The fact that Dublin doesn't have any air passenger taxes like ye have in the UK is another consideration