I just got off the phone with my buddy that is a controller for the Aspen tower/tracon, he said that note was for two reasons:
First, he said basically what boscenter said about the descent profile requiring much more than the 3 degress that the papi will give. If you were to break out right at minimums, DBL 11 DME at 10200, you will still have to descend 2400 feet in 1.4 miles. Because of this, and some other factors (i.e. terrain), all approaches are considered "circling", even if you are able to land on 15 without turning. On windy days, he noted that the faster traffic won't be able to descend as rapidly as is required to land, even on 15, so they will fly somewhat of an overhead pattern, circling over or north of the airport while descending.
The other component that he mentioned relating to this note has to do with the PAPI usable area. If you look in the AFD, or on the Jepp Aspen Facility chart, it says that the papi is only usable within 4 miles of the runway. This generally isn't a problem if you follow the step downs as published to the DBL 9.5 DME (which is within 4 miles), but can be troublesome if you start the approach, then get the airport in sight and cancel your IFR (taking the visual), and follow misleading papi indications from several miles out.
He made another interesting comment about this approach that I don't think I would have picked up on otherwise, and that is the offset localizer that you use in the missed app procedures and in some of the DPs. Even though labeled "back course" on all the charts, it is "normal sensing". In other words, even though you are flying outbound on the localizer, away from the antenna, it does not reverse sense.
Hope all this helps,
Max