This is not a tech problem. It is a design problem.
Good design can't just be "resized". You essentially have to redesign it from the ground up. It takes a significant amount of work. I would rather just use 1024 and have it work on tablets + notebooks + desktops, and then do something separate for mobile. Mobile is such a radically different experience compared to a typical computer you almost have to do it.
It's not necessarily the fact that site is designed to be resized or not it's more about the careful consideration of how a site will look at various sizes. Take http://hicksdesign.co.uk/ for an example. You can tell that the site was designed for various resolutions but uses the same codebase to support them. I don't think the two problems are mutually exclusive.
That is an incredibly simple website. It is possible to get away with it when you have such a simplistic website where the secondary content is almost irrelevant as far as importance go.
Now try and apply this design restriction to something other than a blog where actual interaction is required and it will fall apart fast.
Good design can't just be "resized". You essentially have to redesign it from the ground up. It takes a significant amount of work. I would rather just use 1024 and have it work on tablets + notebooks + desktops, and then do something separate for mobile. Mobile is such a radically different experience compared to a typical computer you almost have to do it.