Good conversation so far. The idea is definitely something we have tossed around, but when we try to determine how to handle this it's a bit more complex.
For a tag such as MediaMath this solution would work well because Tealium is building the call to be made and also firing the image request.
However, tags such as Google Analytics, SiteCatalyst and Criteo have a library that handles building the call and firing the appropriate request - so Tealium is only declaring data and triggering the tag's function so their library can manage making the request. We would not be aware of the call that was made, so that is one issue we run up against.
Also, tags vary in the way that data is sent. Some vendors use the standard query string parameter ? followed by an &. Some use semicolon delimited data though there are several other designs. We would have to build into the tool the ability look at the tag level and ensure we know the expected results to parse. This is quite cumbersome and ideally not the best approach.
I am a firm user in logging tools because not only does it tell you the data sent, it also informs users with call status (404, 200, 302), response scripts (are other tags being piggybacked?), header response, and more.
Hopefully this helps, let me know if you have any follow up questions.