Why is tailwind during final approach and landing so dangerous?
It's not so much a matter of "pilots typically get very uncomfortable" as it is "pilots recognize that it is an inherently less safe situation", and pilots (at least the ones you want to fly with) tend to be somewhat safety obsessed.



So Why are tailwinds during landing "bad"?
The same reason tailwinds in cruise flight are good: you're moving over the ground at a higher speed.
The amount of energy the aircraft has (and thus its landing distance) is roughly proportional to the square of the groundspeed.

Let's consider the plane I fly as an example (a Piper Cherokee) because I know the numbers:
if I'm flying by the book then when the wheels of my plane hit the ground my indicated airspeed should be around 45 knots.


No Wind
If there's no wind my ground speed and my indicated airspeed will be about the same - my brakes will have to slow me from 45 knots to a stop before I run out of runway.
The brakes on the Cherokee aren't Porsche brakes (they're tiny little things), but they're up to the task and a book-perfect landing would have me stopped in 600 feet.
15-Knot Headwind
A 15 knot headwind reduces my groundspeed to 30 knots - at 2/3 the speed I can stop much sooner with the same braking force, which means less wear on the brakes and a greater safety margin of runway remaining.
A book-perfect landing in these conditions would have me stopped in about 270 feet.
15-knot Tailwind
A 15 knot tailwind increases my groundspeed to 60 knots. The tiny little brakes on the Cherokee CAN stop me, but they're going to take much longer to do so: A book-perfect landing would take over 1000 feet to stop (nearly twice the "normal" no-wind landing roll).




The big consideration above is landing distance -- as you noted in your question you can solve that with a longer runway (more pavement means more room to stop), but there are other dangers with tailwind landings, mainly in the fact that Landings don't always go according to plan.
The slower your groundspeed at landing the less energy you have to get rid of if something goes wrong

Consider the three landings I outlined above, but let's add a problem to the mix: One of my tires blows right at touchdown, and the plane veers off the runway into a ditch.
There is much less energy to dissipate with a 15-knot headwind (30kt groundspeed) than there is with a 15-knot tailwind (60kt groundspeed) - it's roughly the difference between crashing your car driving through a school zone versus speeding on a highway.  



Fortunately it's really easy to avoid landing with a tailwind -- most runways are bidirectional (Aspen, which you mentioned in your question, is a notable exception due to geography).

ATC (or at uncontrolled fields, the pilots themselves) will "turn the airport around" once a certain tailwind limit is reached.  Precisely when that happens depends on the airport and the traffic mix (at my home field, with mostly small piston aircraft, ATC will typically switch the runway when the tailwind component exceeds 5 knots - at JFK they might land a jet with a 10-knot tailwind rather than messing up the arrivals).
Of course there's nothing that says a pilot has to accept a landing with a tailwind: If the pilot is not comfortable with the conditions they can request a runway aligned so the winds are more favorable. (There is a somewhat famous case of an airline pilot at JFK who ultimately decided to declare an emergency to ensure a landing on a favorable runway).