I'm far more a phoenix stance player than Tiger, but I'll agree on the general idea of maxing one over the other, and not ignoring bear stance.
My biggest argument for Bear over Tiger would be cost/benefit style. Juggling 4 stances is fine and all, but early game you're not going to have the mana to sustain such a thing. Ever since the patch-changes, maxing your stances lowers your mana costs for them, meaning you can spam those moves more often and not be as reliant on your mighty Blue buff.
Taking the Utility-Udyr build's CDR masteries, the moment you hit blue you'll be able to spam 1 stance over and over without using up your passive, It's amazing to me how much that CDR works for Udyr. Then I throw in the movementspeed masteries and runes, and I've got a level 1 Udyr at 372.
Bear stance being what it is, it's extremely important for initiating ganks and escaping enemy fights. The lower the mana and longer the run on Bear the more likely you'll use it to Travel earlier as well. Don't be a dork at 100% Mana not using your Bear to get around, nobody likes a late-to-the-party Udyr. I tend to Max Phoenix into Turtle, with Bear for the 3rd.Once I hit Phoenix and Turtles max I'll likely have my Golem+Swiftness combo and be able to utilize the bonus movement speed more and more.
Leveling Tiger instead of bear means you'll want to spend more points in Turtle, and you'll not have Udyr's one CC. A Rank 1 Bear is fine, but a Rank 1 Tiger is unacceptable. You'll be more inclined to stay inside for duels and not utilize your Phoenix stance (Since you'll go this if you're arguing over Tiger/Bear) which is far stronger early, and already has more points in it.
Udyr doesn't "require" Tiger to push a lane, his amazing lane clear and 30% ATS steroid already do it, an additional 20% won't hurt sure, but you'd be better off being able to escape than shaving 30 seconds off the kill timer.
I'll disagree having your movement speed kills dueling power. Udyr's early dueling power is exceptional, and while Phoenix might not beat Tiger 1v1 while maxing his Bear, he'll sure as hell gank faster, clear faster, be in more team fights, and be kited during far fewer fights. Bear/Phoenix is more about maxing your early game potential, and the faster the Udyr the more he's able to do.
Before the changes I'd never suggest leveling Bear beyond 3, with the short and scaling timer, stronger % bonuses, and the huge mana-reduction costs, it's far more reasonable to level Bear than it used to be.