You raise a good point. I don't think that giving us control over the character right after character creation is a good idea. Cutscenes establish the starting point, set up the scene and only then you get control of the character to interact within the scene. Removing cutscenes will probably be quite confusing for a player and you want to draw them in right from the start.
I do agree about having an ability to view the skill tree prior to finishing the character creation. Not having it results in a slight immersion break when your character is put in an urgent situation (ME2 and ME3) but you spend time leveling up.
"You don't have time to wait around, Shepard!"
"Sorry Miranda, I need to level up first" 
I would even go farther and give players the ability to spend points in the skills and choose their own loadouts. Consider, an Engineer in ME3 starts with a point in Incinerate and Combat Drone (and a bonus power, if applicable) while the other skills are unlocked when you reach a certain level. If your build does not involve Incinerate or Combat Drone you are still forced to play through the prologue and Mars missions with "wasted" points. Only then you get access to the Normandy and can reset Shepard's powers. It's worse in Mass Effect 2, when you are forced to level those powers regardless if you want to get access to the others. Not to mention that you have to play through the Lazarus station, Freedom's progress and Mordin's recruitment mission to get access to the research lab. In Mass Effect 1 there is no possibility to reset powers altogether 
So what do I suggest? The next step after picking a class and bonus power should be the "Squad" screen. Player gets to spend two points (three, if a bonus power is available) to pick the initial powers. Other powers are level-locked afterwards. There will be a "Default values" button that creates the predefined class. After picking the skills, weapon loadout is shown (with at least one weapon available for each slot) and the player chooses his/her equipment. Only after that the cutscenes start.
So, for the Engineer example, here is what it looks like in ME3:
Incinerate 1 point
Overload Unlocked at Level 3
Cryo Blast Unlocked at Level 6
Combat Drone 1 point
Sabotage Unlocked at Level 3
Sentry Turret Unlocked at Level 6
Tech Mastery
Fitness Unlocked at Level 3
Bonus Power 1 point
Here's my suggestion
Initial screen (3 points available)
Incinerate
Overload
Cryo Blast
Combat Drone
Sabotage
Sentry Turret
Tech Mastery
Fitness
Bonus Power
After spending points in Overload, Sabotage and Tech Mastery
Incinerate Unlocked at Level 3
Overload 1 point
Cryo Blast Unlocked at Level 6
Combat Drone Unlocked at Level 3
Sabotage 1 point
Sentry Turret Unlocked at Level 6
Tech Mastery 1 point
Fitness Unlocked at Level 3
Bonus Power
Bonus Power is never level-locked.
Or you can spend those three points in one power (even a non-combat one, in that case you'll be forced to rely on your weapon instead and level locks will be applied by default).
Incinerate
Overload Unlocked at Level 3
Cryo Blast Unlocked at Level 6
Combat Drone
Sabotage Unlocked at Level 3
Sentry Turret Unlocked at Level 6
Tech Mastery 3 points
Fitness Unlocked at Level 3
Bonus Power