Your users care less about privacy than you think.
SO MANY analytics posts here on IH inevitably veer in the direction of privacy. Most commonly, there's a perception that using Google Analytics is "feeding the Google monster", or that using analytics tools is a violation of privacy.
I have a heretical opinion here: your privacy preferences are not those of your users.
My day job, in which I see the opt-in rates for 100 million users, has convinced me of this. I am not at liberty to disclose precise numbers, but they are beyond sufficient to convince me in a personal capacity that most consumer bases are less concerned than you are.
I think this is one of the biggest misconceptions when it comes to analytics on IH. This community loves privacy, which is great. But members of this community also project that preference on to their users and on to consumers in general, which is a logical fallacy.
If you and I launch competing products, and you refuse to use analytics while I do, I assure you that I will crush you in competition.
Privacy is a worthwhile endeavor, and certainly there are tools and technologies that are "creepy", or overreaching. But there are plenty of valid use cases that don't infringe on privacy while providing you information to improve your business.
Before commenting, consider having a read of a longer form blog post: https://analyticsforindies.com/your-users-care-less-about-privacy-than-you-think/
This is a good read, and I agree there's a middle ground between "collecting no analytics" and "full–on creep mode". And everyone has to figure out where the line is for themselves (although I think some technologies — like session replay, which you mention — are inarguably bad).
One consideration that you don't mention is what the analytics company you use does with the data. For example, Google Analytics feeds into Google's pervasive tracking of me everywhere on the Internet. So even if you're not using it to be creepy, there's a negative externality that's not present with a more ethical analytics service like Fathom.
(Also, "it gives you a competitive advantage!" is an argument that can be used in support of basically any unethical business practice. So too do sweatshop labor, covering up workplace misconduct and using tax shelters, but that doesn't mean they're okay things to do.)
Ultimately, I choose privacy–conscious analytics services for the same reason I use GitLab and DigitalOcean over GitHub and AWS. My users may not care that my infrastructure provider doesn't e.g. help ICE deport undocumented people, but I do. I don't make that choice so I can market myself as ethical — I make it because it's the right thing to do.
I care a great deal about privacy myself and accept that most users don't. My problem with having Google Analytics on my site is not that users might be turned away, but rather that I am facilitating tracking users' behavior by Google, through my website, meaning I would help doing the one thing I hate and I absolutely refuse to do that.
My take is not not using analytics, far from that. I will probably use something self-hosted or that has a more privacy oriented approach.