Can I play that mobile character in core? No, not really. Your fighter with the same feats can only be roleplayed slightly differently, and those intstances of differences won't be reflected in combat style. It's because I don't view my character as just a collection of attacks/defenses/stats that I am capable of pulling off different characters, allowing for that guy who likes to move around and hide, or that guy who punches you for negative energy damage, or the Incredible Hulk (changeling psychic warrior/warshaper/monk, it's fun).
Those roleplaying differences won't be very big. The difference between ordering lager or ale at a bar, for instance. That's just the same guy, dressed in a different outfit. That's repetition. If you like playing the same character over and over again, by all means, restrict things to core. If you feel you want to try new characters occasionally, there's a vast resource of great classes and PrCs out there.
On the subject of core balance/imbalance: Core you have fighters, monks, and paladins along side clerics, wizards, and druids. That's three tier 5 classes with three tier 1 classes. Of your 11 classes. The others are barbarian, ranger, bard, rogue, and sorcerer. The rogue is less powerful in core, due to sneak attack woes, it's probably tier 4 or 5, same with the ranger and barbarian. Bard and sorcerer are up there in tiers 1, 2, and 3, probably just 1 and 2 core only, due to the still smallish amount of spells and the still hilariousness of diplomacy and bardic music effects. Do you see the massive gap here in class power? Bring in stuff outside of core, and the tier 1s get more powerful. Do to access to more stuff. But the lower classes get a tremendous boost in power here. Rogues now have the means to SA some creatures, or all creatures, and settle into their tier 4 position. Bards move to tier 3. Monks now have the ability to take Fist of the Forest and instantly fix some of their problems. Paladins have CC. Fighters can now do lock-down effectively. Barbarians have charger builds. The list goes on. Sure it does give a boost to the Big 5 (6, erudite with spell to power variant is currently under discussion), but it's not game breaking at all. Overall it just gives you more choices.
Now, I'm wondering how many game breaking things are actually core. Wish/miracle. Those can be abused in so many ways. Fabricate/Wall of Stone/Iron. Diplomacy. That's three game breaking things right there. Polymorph line. The Planar Binding spells. All of these are immensely abusable. Out of core? I can't think of any right now.
Also, well put NiN.