Am I doing something wrong?
No I don't think you are.
If you want to preview your remote site then you do that in the Browser [i.e. by navigating to the URL]. I think that most people will use the preview of the local site to compare with the remote site. Well, I do anyway. I also think that what you are seeing is what DW intends Preview in Browser to achieve.
Thanks. I'm used to Coda which has a preview for remote and a preview for local.
Is there another quick way of getting to a live view of the file on the remote server?
Well for me it's still a bit manual! I just keep my Browser open on my page and alt-tab between them and f5 refresh.
There may be better ways that more experienced DW users come up with!
That might be hard to get used to.