Yep. By all means you could play as one of the ordinaries. There's nothing stopping you from doing that, but you're cutting out a large chunk of the gameplay by not including the powers related to the Major Arcana cards. Fateweaver being a TTRPG allows you to play whatever you really feel like :) A political intrigue among the Elders, goody-goody adventure to save some poor Ordinaries, or some dungeon delve to uncover hidden secrets of the Fateweaving itself :D
As for choosing your affinity as a commoner, you could still do it but this would be more related to your job in the society. A farmer would be a sword, needing physical strength, while a scientist could be a wand for their smarts. It's all fully playable as ordinaries where Fateweaving is something you heard of but never truly understood nor seen. You could play a whole group of ordinaries who are hoping to challenge the status quo and refuse their pre-ordained life status :D Plenty of ideas just from that bit itself I'd say!
Have fun and make it your own!