Sharpshooter41,
I am not familiar with the actual day that you are referring too, however here a few points that may, or may not, be relevant:
* Did you advise ATC upon arrival of the tailwind?
Did other aircraft experience and report the same conditions? ATC rarely "swap ends" on a single report unless it is blindingly obvious of the need to change. Its normally more subtle than that and you have to anticipate or identify a trend.
* Were you advised of the tailwind by either the ATIS or ATC either prior to or during your approach?
* What was the traffic situation? Was it a busy arrivals sequence? It normally takes at least 1/2 an hour to "swig the ship" during a busy arrivals sequence. Trying to balance avoiding holding and getting the last departures out can be a dark art and unfortunately you always have the last to arrive guys that invariably get the worst of the weather that has instigated the change.
* If it was unsafe why didn't you conduct a missed approach?
* What was the wind at the other end of the runway? It is more common than you think for the wind not to "play fair" and ATC have operate with the lessor of two evils depending on RWY's available and configuration.
* In my experience exit TWY's are normally requests - rarely requirements. We learn from experience of watching countless arrivals and departures that there are too many variables at play. You can be pretty sure, but there is always one that floats of doesn't brake like the rest etc etc. And besides you are PIC. If you can't make the requested exit, let ATC know and we move on to plan B. Or C. Problem solved.
I know this doesn't answer your specific question, but what I am saying is that there is often more going on than you perceive.