I'm not sure that's the reason. I think it has more to do with the dragon automatically winning. At that level difference, the dragon is less of a foe and more of a plot device.
Yes, automatically winning is the thing. Why let the fight last 3 rounds?
Let me rephrase. The 4E mechanic doesn't break down, since a level 15 solo vs a level 3 party could still work, technically, in that you could run the combat between the dragon and the level 3 party and the dragon would still kick their butts and everyone would still roll dice and the "rules" would be followed. The 4E mechanic is however not balanced for allowing a third level party to encounter a level 15 solo and make any sense.
The warning in the DMG about avoiding opponents that have a too important a level difference with the PCs is clear and it was not made arbitrarily. It was made for a specific reason. It's not just that the monsters will be too strong (or too weak). It's that the game is not built for such encounters.
For example, the game provides guidelines for creating monsters that will be balanced at a certain party level. If you want a monster that will be clearly overpowering like a dragon that would kill PCs in a single round, I don't see what prevents you from creating one.
Now I understand that the question is rather: but why doesn't the level 15 solo monster work out of the book for killing the level 3 fighter in a single round? As mentioned above, out-of-the-book means you apply the book rules and the book rules mean no level 15 solo against a level 3 party, for the reasons set out above. I don't see the point in arguing that you could do it; of course you could. The point is, that an adult red dragon is wayyyy too strong for a level 3 party, as mentioned by the OP. And to convey that wayyy-too-strongedness, I believe it requires that you simply deal damage which happens to be more than the fighter's HP with the dragon's breath weapon if that suits your fancy. If you're setting up a fight between a level 15 solo against a level 3 party, the intent is obviously to wipe out the level 3 party anway (if they can't flee), I see no reason to let the agony last more than a single round. By using a level 15 solo against the level 3 party, you're not following the rules anyway if that happens to be important to you, so you might as well tweak those dragon stats so that they make sense for a level 3 party.
Sky