Sounds like a great opportunity for someone to sell some old-fashioned on premise software.

On-premise is not old fashioned. It is indeed a requirement in several countries, for many domains including finance, banking, healthcare, telco and such. When you include PII in your analytics service, it has to be taken care of explicitly. There are indeed a few on-premise mobile / web analytics platforms for product analytics purposes that you can deploy on your own servers and retain your users' data with your own rules with no dependence on 3rd parties (disclaimer: I am cofounder of Countly).

Just some minor pedantry - that should be on premises. Although it's pretty commonly misused, technically premises should always be plural when referring to a property/residence/place of business, and premise (as a noun) only means "a previous statement or proposition from which another is inferred or follows as a conclusion."

Already exists, it's called Matomo and it is awesome.