I know AWS in particular does not because they do not increment the bill for every request. I don't know exactly how they calculate billing, but based on what I do know about it, I imagine it as a MapReduce job that runs on Lambda logs every so often to calculate what to bill each user for the preceding time interval.

That billing strategy makes it impossible to prevent cost overruns because by the time the system knows your account exceeded the budget you set, the system has already given out $20k worth of gigabyte-seconds of RAM to serve requests.

I think most other serverless providers work the same way. In practice, you would prevent such high traffic spikes with rate limiting in your AWS API Gateway or equivalent to limit the amount of cost you could accumulate in the time it takes you to receive a notification and decide on a course of action.