I think that is the key here, Internet media helped them disseminate their message very rapidly.
I would also argue, easy access to capital was very important. Once that original video went viral, they had to really pour on the online ad-spend to acquire customers. Seeing as how their customer acquisition costs are high, the more customers they got, the more money they lost every month! Capital is key!
They were wildly successful before they were on AWS. I believe they started off on magento, then drupal before writing their own software (once they were having a lot of success).
They don't use Amazon for fulfillment. They used some other 3PL's and are now leasing and running their own distribution centers. In the beginning, Dubin was packing those razors up in his living room!
Source: founder a close personal friend.