I think the two biggest misconceptions about balance (with repsect to PCs vs encounters, anyway) are:
1. The PCs always win a balanced encounter.
2. The PCs always have balanced encounters.
The PCs may be
likely to win a balanced encounter, but as with any game of skill and chance, the players may make tactical mistakes and the dice may not always go their way. Even if the PCs have a 90% chance of winning a "balanced" encounter and only ever fight "balanced" encounters, that means about one in ten encounters will end in defeat.
Furthermore, whether or not the PCs will have balanced encounters is entirely up to the DM in a non-sandbox game. In addition, encounter difficulty is not a binary "balanced"/"unbalanced" switch. There is a continuum of encounter difficulty, from encounters that the PCs are almost certain to win, to encounters where they have an better than average chance of victory, to encounters which could go either way, to encounters that the PCs would do well to run away as fast and as far as they can.
Encounter balance is descriptive, not prescriptive.
So in the actual game, how do you run #2?
Put them in a situation where they face a hard encounter; according to the DMG, one of Level+2 to Level+4.
If you use monsters that are genuinely stronger than the PCs, like the criminal in your story, how often does it end like your story, and how often does it end with the heroes making a valiant effort, but in vain?
In a story, the author chooses whatever the outcome he wants, however unlikely. In a dice-based game, the outcomes will be dictated by statistics. Heroes facing truly unlikely odds will be truly unlikely to succeed.
From a certain perspective, facing truly unlikely odds and actually succeeding is what
makes them heroes.
However, you've put your finger on the key difference betwen a game and a narrative. In a narrative, last, desperate, one in a million chances might come up nine times out of ten.

In a game, nine hundred and ninety-nine thousand, nine hundred and ninety-nine times, the PCs lose.
Some games offset this by giving the players narrative tools which make that one in a million chance more likely to come up: in D&D, these include action points, second wind, daily abilities, etc. However, this shifts the
actual chances of the PCs' victory back to something closer to 50-50, which by the previous definition no longer makes them heroes.
As for me, I base my definition of heroism not on beating the odds, but on doing the right thing. In my games, you can be a hero by doing the right thing even if you have an 80% chance of success. If the PCs are in the game long enough, the dice will make them fail often enough anyway.