Donate to your local library.
This is the entire point and purpose of the public library system: aggregating resources to provide accessible information and entertainment for local communities.
Your local library almost certainly has subscriptions to several online and physical news sources. Typically larger libraries have physical newspaper subscriptions from across the country.
If you want to improve accessibility for everyone while still keeping publishers paid, donate to your local library and tell everyone you know about the services they offer. Seriously, we already have a nationwide system to provide access to information that not everyone can afford. We don't need to invent something new and more complicated.
It’s a nice sentiment (librarianship runs in my family) but there are still issues with accessibility for library institutional subscriptions, compared to easily shared links. Relying on institutional subscriptions for revenue isn’t enough for the publishers and journalists. It’s also hard to target donations towards certain parts of the collection; at most libraries almost none of your donation would go towards periodicals.
None of these problems are unsolvable, but typical libraries would need to significantly change their focus and invest in new shared technology first.
I do recommend contacting circulation, reference or technology direction to share your desire to see the library play a larger role in making journalism accessible to the community.