There should be an intermediate syndicate that charges me micropayments for every article I choose to read, then charges one lump sum to my credit card at the end of the month. And also remits payment to each newspaper or Website.
You fill a bucket. It's drained, based on what you read/view/listen. Or otherwise equitably shared based on some global allocation basis if access nothing --- you're still benefiting by the positive externality of the informed polity which journalism creates --- if you read nothing.
This ensures a stable funding basis, you have a predictable cost basis, you can direct the allocation based on your own access patterns, the common weal benefits even if you don't utilise the resource.
Note that much of this is the same as an ad-funded media, excepting that you can't direct spending, the allocations are far less public-benefit oriented, and the costs per household are far higher: roughly $700 per person for advanced countries (North America, Europe, Japan, Australia/NZ), based on a $700 billion spend and roughly 1 billion population. What we have now costs an immense amount and is failing media and journalism badly.