Can you play the main story section as a standard street fighter or does it have to be custom?
I would get more fun out of sticking to one character and progressing instead of a broken build I would need to eventually discard.
You can do that, its a custom character sure but you can just stick with one of the masters and use their moves only
It must be a custom character. Across the story mode you unlock as possible masters the normal characters of the game. You choose which ones are your masters.
Once you become apprentice of one of them, you unlock their fighting style, meaning you unlock their basic animations (intro, victory pose, idle...) and normal attacks or basic grabs, plus a couple special attacks.
Then depending how much you use this style (which increases your style rank for this master) or how do you increase your bond with the master (via giving him/her gifts), you keep learning / unlocking new moves from that master.
Each master gets unlocked in a different point of the story mode: the first one is Luke, the one unlocked just before the end is Juri, and the only one unlocked just after completing the main story of the World Tour, starting the endgame is JP.
So to use the moves of your favorite character, or better, mixing your favorite moves from different characters, you'll have to unlock and take them as masters and then to upgrade their style rank/bond to unlock the special attacks or super arts you want.
As a reference, the level 3 super art gets unlocked when you max out at level 20 the style rank with that master. Also, the stuff like your Super art levels or the Drive meter bars get unlocked as you play the World Tour mode, not sure if when reaching certain level with your character or at certain points of the story.