The PKI system was designed independently of the web and the web used to be one usecase of it. You're kind of turning that around here.
"WebPKI" is the term used to refer to the PKI used by the web, with root stores managed by the browsers. Let's Encrypt is a WebPKI CA.
You're trying to make it sound like there has ever been some kind of an universal PKI that can be used for everything and without any issues.
WebPKI is the name of a specific PKI system, where PKI us a generic term for any PKI.