As I've posted many times, we are a pretty successful UK-based ecom, and we are not price competitive.
Our USP is not in any way about price. We tend to be 2-5% more than Amazon, and 10% more than the "bottom feeders" who drop ship.
If you think the only way to sell is being the cheapest, I think you are bound to fail.
Edit to add-
Actually, if you are selling on price, you DO NOT need a brand. What you need is non-Brand trust signals the let people FEEL that it's ok to buy from you.
If your target is bargain hunters, you need to keep your overheads low, your profit margins razor thin (i.e. be cheapest, not just cheap), and you need to give people "permission to buy". And you have to hope someone doesn't come along who is cheaper than you, or all is lost.
To me, that is a high-risk business model that I would not be comfortable depending on. You could EASILY be taken out by any big company who buys in bigger bulk than you and gets lower prices. Or any company who has the cash reserves to run at a loss to kill the competition, then hike the prices later.