The point is that this could be solved by a solution that is completely external to their current infrastructure, meaning they don't even have to mess with any integration or legacy code. Agile in this case seems perfectly suitable because they can easily have a fallback when the server doesn't respond to a "time"-request, that falls back to the current state (i.e. use client-side time). So we've got the perfect storm of no integration with current systems, tiny overhead on requests (it's basically an ECHO server), unimportant feature and easy fallback.

In fact I have an enterprise plan to sell them that does exactly this if they would like to externalise this problem.