I regularly use my bluetooth mechanical keyboard (nuphy air 75) with both ipad and iphone. Works fine. Much better than the keyboards sold by apple. What are you missing?
My point was that there will never be a full “desktop mode” or even a good generic (supporting all apps) way of running apps on an external display. Right now you can develop apps that draw on an external display, but that has to be done on a per-app basis (there was a notable one a few years back that used “web apps” to build a “desktop” on the external display, but it’s not full, generic, zero code changes native app support).