Absolutely not. They can't gain sales from this because they're sold out. High opportunity cost - flatline zero revenue for the next week, which is (at least one of) the highest demand weeks of the year.
2) I bet that the average cart size at checkout is less than half of the cost of an Echo.
The benefits don't outweigh the costs of a marketing stunt of claiming that they are sold out.
Birthday is different though, but will not be the case for most decisions.
Long term gains > Short term gains on a product like this.