1. Government subsidizes media. No cookies or profiling required. However, unclear if one government should support articles that are mostly read abroad and, of course, there are many issues with this related to equitable distribution of funds and impartiality.
2. They can run ads. However, ads tend have better ROI when well targeted which requires profiling. For instance, if I am reading an article about suits. If they know my age they can make sure to show me people around my age wearing that suit. Which makes me more likely to envision myself wearing it and thus buy it. Cookies (or similar) required.
3. A paywall where the reader pays for it directly. However, we seem to be very resistant to that idea given we were trained to use a "free" (sell your data model) internet. Nonetheless, by seeing your browsing history from your account server-side (no analytics cookies needed) they can place you in cohorts server side. So not that much more private. If anything, less private since they have your real data through your payment method.
Ensuring privacy while still getting access to free things and not bankrupting companies (especially small ones) is a hard problem.