But I’m worried doing so will penalise the website in Google results.
My searches led me to direct answers from Google staff that no, they don’t take Google Analytics into account in rankings[1]. However, those answers are a decade old, from a time Google still touted “don’t be evil” as a motto. The Google of 2022 is one I don‘t expect to keep promises from 2010.
Are there reliable and/or recent sources which elucidate the case either way? Anecdotal reports can also be helpful.
[1]: https://plausible.io/blog/google-analytics-seo
Google collects all they need and more from users using Chrome. They don't need GA anymore, and I'm pretty sure they said a while ago they were going to stop looking at the data (again, because they collect far more data from user using Chrome).
Good on you!
John Mueller is a Search Advocate at Google and has weekly-ish Youtube office hours where you can ask in greater detail, if you'd like.
Example - Wikipedia, Facebook, Apple and many more. They didn't have GA.
https://usefathom.com/blog/google-analytics-seo
They argue that it's better (?) because your website will be faster.
"This is because Fathom’s tracking code is super-lightweight and blazingly-fast to load. Our tracker file is tiny whilst Google Analytics’ file is much larger. This is a little funny, because Google’s search engine rewards faster sites, so a case could be made that using Fathom instead of Google for your analytics could potentially increase your page rank in their own search engine!"
A counter argument would be if you use Google Analytics for more advanced tracking of user features, such as A/B testing & tracking how users interact with your site.
Thats not how search engine optimisation works. Neither in theory nor in the real world.