i think your synopsis is great! very insightful. i came up with similar conclusions, but i was left still wondering about braids true meaning. But after reading this, Im almost positive that this is the game true interpretation. though i think you have might have been reaching a bit with the whole
"the most he can do is concoct a cartoony saving-the-princess scenario based on the video games he played as a child."
All though it would explain all the homages to mario bros and at same time pay homage to the genre in general.