"Of course the best approach would be for the site to check for required browser features and only pop up a warning message if a feature used by the site is reported as not supported by the user's browser."
A better approach still would be a plain vanilla test browser approved by someone such as the W3C. Devs could use it for their own testing but if it passes W3C or whoever could also hand out an approved badge based on independent testing. Banks and other financial institutions would be required to have it.
Lack of the badge would become an indication of cowboys and smartarses at work.