No, Urban barbarian is not capable of using a normal rage.
This was clarified by James Jacobs on Paizo's official forums.
It does indeed replace normal rage. An urban barbarian doesn't get to apply a bonus to both Str and Con. He instead gets only one +4 bonus, but has the advantage of being able to choose Dex if he wants or of being able to split the bonus up how he wants.
If you're an urban barbarian, you can't rage like the normal barbarian. Because Controlled Rage IS your rage. It doesn't make sense that you'd be able to do a normal (uncontrolled) rage when you're playing an archetype that is all about controlling the rage.
So that means the "may" in this case refers to the choice of which stats you want to increase: Strength, Dexterity, or Charisma, and not which kind of rage you get.
No, there is no good balance reason
The designers may have imagined there was; it’s conceivable they considered the option of Weapon Finesse as your 1st-level feat (or Weapon Focus, Exotic Weapon Proficiency, Quick Draw, and so on) to be important advantages of full-BAB classes. I tend to believe it was more of an “imagery” thing, though – full-BAB classes were the “weapon masters” and so weapon-related feats tended to get this requirement. Not because balance demanded this, but rather because their image of what each class was “supposed” to be like. But either option is pure speculation on my part, nothing more.
The theory:
Regardless of what they were or were not thinking when they wrote the feat, the BAB +1 requirement on Weapon Finesse is not only unnecessary (just as it is not for Weapon Focus, Exotic Weapon Proficiency, or Quick Draw), but is also exceptionally problematic. That’s because, unlike the other feats I mention (with the possible exception of Quick Draw), Weapon Finesse is a build-defining feat – someone with Weapon Finesse is going to have low Strength and high Dexterity, and is going to be interested in melee combat. Thus, they need Weapon Finesse to function correctly.
Worse, this description, of a high-Dexterity, low-Strength melee combatant, does not describe most full-BAB classes – you can make a Dex-based fighter, certainly, and even a Dex-based barbarian with the right supplemental material, but these are not the norms for those classes. Fighters get proficiency with heavy armor and shields as a major class feature; you cannot use those on a Dex-based build. Paladins are worse – same proficiencies, but far fewer feats. And rangers are worst of all – their ability to ignore the Dexterity requirements of their Combat Style feats is a major class feature, one of the only ones that is both relatively unique and relatively useful for that class.
And then you have medium-BAB classes that, unlike the full-BAB classes, are often perfectly described as high-Dex, low-Str melee combatants. Monks and rogues wear light armor if they wear armor at all, have ¾ BAB, and find themselves in melee. Rogues in particular see a lot of advantage from high Dexterity: requirements for Two-Weapon Fighting feats, associated score for critical skills. These are characters who want to build around Weapon Finesse.
Which sucks if you are level 1, and have BAB +0. You are forced to sit through two whole levels of hideously poor accuracy, just so you can make the math work by taking Weapon Finesse at level 3. This doesn’t add anything to the game, it just makes those classes suffer needlessly.
The practice:
The BAB +1 requirement on Weapon Finesse (and most other feats, honestly) is waived in the overwhelming majority of games I play in (the only games where it isn’t, that I can recall, are those in which no one wanted it and therefore didn’t bother asking the DM), and every single game I DM. In fact, most of the people I play with forget that it has that requirement. And our games are better for it. It allows us to play at low levels without being forced to suffer through unnecessary glitches like those caused by Weapon Finesse’s requirement.
I strongly recommend every DM to waive that requirement. I’d also recommend doing so on other, similar feats (certainly each of the ones I listed earlier). To not do so, I think, is a mistake. A mistake that the authors of Player’s Handbook made, true, so an understandable mistake, but a mistake nonetheless.
In fact, I would echo Thales Sarczuk’s suggestion, and strongly consider just making Weapon Finesse a free option.
Best Answer
There is no longer a trade-off of defense vs offense
By allowing rage/barb features to work with Dex then the Barbarian becomes much less MAD when having to decide point attribution for Offense vs Defense.
In the official scenario, you have to decide between CON/DEX for higher AC/HP or STR for better attacks. If you can lump them all together, then the Barbarian gets much better AC and still gets strong attack options.
What you lose by not pursuing STR
Weapon Types/Damage dice
The biggest loss is the removal of heavy weapons or the largest damage dice. But that's not an enormous deal as you can get a 1d8. but you lose out on heavy property items/feats/mechanics. But that's not anywhere comparable to what you gain.
Saves
As with all ability scores, there are often saves associated with them. Lower STR means more likely to succumb to STR based mechanics.
Carrying Capacity
Many tables hand-wave this, but for those that don't, a lower a strength means you carry less.
Grappling
While you can use Dex(Acrobatics) to contest a grapple, you are still required to use STR(athletics) to initiate one. With a lower strength, this special melee attack becomes a less likely option.