No they don't. CDNs with the exception of AWS Cloudfront don't run on a cloud provider. Things aren't quickly spun up and down with any kind of velocity. CDNs manage all of their own infrastructure, POPs, hardware etc. There is usually a maximal hardware lifecycle since the CDN business is so CapEx heavy.

Cache-Control TTLs and DNS SOA record TTLs have nothing to do with each other. They are separate layer 7 concerns. The former is used by your browser and the latter by your system's resolver library.

I have never heard of anybody using SOA TTLs records to purge a cache. How would that even work?

CDNs generally give you decent tools to purge your objects.

Source: I worked for a CDN.