Advantages of using context for sitewide blocks?
Yes you should. It is worth while to switch to Contexts (Irrespective of your case about site wide context blocks)

Reasons


When you have blocks which appear only on a certain pages, the time taken for loading the block is saved.
In one page you can have a block in region A and on other page you can have the same block in region B.
When you change your themes you need not worry a lot about changing all the settings. You just need to make sure that they have common regions.
You can think on a page by page and then assign blocks.
Context has a simple and clean UI of its own, including drag and drop facility.
It is not path dependent. It means you can use conditions other than the path to set a context.
Can be extended with Context UI which can be a replacement for Panels and light weight too :)
In addition to placing Blocks there are many more reactions as well.
Contexts can be stacked meaning there can be multiple contexts for a single page. So you can have a site-wide context to place the headers and footers, then a node context which will perform something only on node pages and then one more context which will be active only on say node/33.
In short think of it like Rules module for the Layouts.