Why does gov.uk, a site all about allowing the British public to interact with the British government, use google analytics?
You are shipping all the data about all my interactions with my government off to a third party in another country. Another country that we know has not got the same legal data protection requirements, and one which has now been exposed as having massive internal spying problems.
And no, telling me "google aren't allowed to use the data" and then opening an outsourced helpdesk ticket with another US based company does not cut it.
Assuming they listen to your suggestion and act on it as you suggest, it seems the only option open to them is to design their own in-house (In UK for that matter) version of Google Analytics to do their own analysis. Regardless of the cost and time this would add to the project, it's unlikely that it would be anywhere as good as Google's offering.
The other, more likely, option would be to decide it's too expensive to implement a different, more complicated, solution; so they don't bother. They don't get the feedback and analysis on how to improve their services and the customer experience declines until you're back where you started with a poorly designed product offering hard-to-find information and people are posting angry comments on HackerNews about how bad gov.uk is and how they would never run a start-up like that... I'm almost certain someone would say "Why don't they use google analytics to improve things, like everyone else".
Instead, we need to be applauding a massive operation like Gov.uk for taking a dose of reality and thinking, "we're not doing anything amazingly special here, we're providing people with a quick way to check their council tax, or bin collection dates, or maybe pay their car tax. let's just get the job done as best we can."
They could just install Mint on their own servers. Problem solved.