Not that I’m aware of. There are several commercial services but there are real costs to running that much testing and getting it for free is not likely. The public WebPageTest API limit is right around where you’d need (200 URLs loaded per day) but you’re looking for a lot of other functionality as well.
If monitoring the performance is important to you, consider using a commercial service where you actually get guarantees and SLAs.
It’s been a few years since the list of 3 above was posted, and it’s a bit outdated.
Here’s the most recent list of speed monitoring services - who still are not free since as Patrick said it costs lots of server resources running these tests on a recurring basis:
I hope you don’t mind that I included ourselves on there - MachMetrics has been around for more than a year and are currently providing automated speed tests to over 400 users, from 5 regions, various devices, multiple times per day - all powered by the wonderful WPT engine