Linux distros log a TON about what you’re doing by default.
Tonnes of software uses systemd-journald to log errors, the bash shell saves everything you type into the terminal, and wtmp, btmp, utmp all track exactly who is logged in and when, and the kernel uses dmesg to log a bunch of stuff.
While the system isn’t sending these logs to Microsoft or Google, anyone who gets into your system like police or hackers can see almost everything that you have been doing.
If you want to be private, you must disable them.
It sounds like you want something that doesn’t lend itself to troubleshooting or useability. It can be done, though: pull your hard drive and run everything from a live USB with no persistent storage. Or just don’t use computers.