So I ventured to fork for the first time the GitExtensions repository and, within a couple of hours, I had a basic integration done thanks to the very useful TeamCitySharp and Reactive Extensions libraries. Here is what it looks like:
I've added a column or the right to display the summary from TeamCity, along with a new tab on the bottom to display the full page from TeamCity concerning the selected build. In case you are interested in trying it for yourself, you can build the sources and even the installers from here.
The architecture I put in place is pluggable, and there are plans to extend this solution to include Team Foundation Server in the future.