Yes. All surfing is between your client and the server running the web services.

The problem is that many many companies/developers used cloud services to host their websites. It's where they out their computers/services that determine the reliance on the big hosting platforms or not. Your client really has no say in the matter.

I use the Internet all the time without cloud platforms, but it's because I host tools on non cloud systems. Having Cloudflare or AWS down doesn't affect my own tools because I'm not hosting them on those platforms.