I think that playing as a non-biotic is kinda inspiring... Makes Commander Shepard a potential member of the Badass Normal trope.
I do believe that perhaps some Shepards are just better biotics than others, though. Some are just better with guns.
As for which class fits Shepard best, there's none to me; the point of Shepard is that the game gives you a template for a character, and you build on that. My favorite class is Infiltrator, but I've also beat the game as a Sentinel (that was fun, too), and I'm working on Adept. I've played as all six classes though. None beat infiltrator for me.
All the classes appeal to Shepard, really.
Soldier - Makes Shepard the badass runner and gunner guy that can fire any weapon in existence.
Infiltrator - Fits well with the Spectre Status, and who wouldn't want a precise sniper leading the galaxy?
Sentinel - Unique tech/biotic mixed abilities make him a powerful combatant.
Adept - Being a powerful biotic makes him a force to be reckoned with.
Vanguard - Isn't afraid to put himself in the line of fire if it means winning the battle in a flash.
Engineer - Appeals to the hacking and bypassing minigames, and hey, he can fix his own gun