They really should. The time to do this is now.
I don't really know why this did not happen already. The only answer I can think of is lack of people with a vision in the management.
Mozilla could leverage a user base of 100 million people. But it's an enourmous challenge to build the bridge between readers and publishers. When you start with the publishers, no one will want to participate because there are no users. When you start with the users, no one will pay because there is no content. So you have to slowly create a momentum.
Another problem is that people will probably never pay as much as ads are worth - people don't see that they pay for ads when they buy new shoes for 200 $, but paying only 40$ directly feels like a loss. But the system is extremely inefficient, so middlemen siphon off most of the value (Google, Facebook). Firefox is also in the unique position of being a potential middle man. I think there are enough people who would be happy to support it with a small amount in the beginning, if it feels promising and rewarding.
So especially at the beginning users need to get something in return that motivates them. Mozilla could start with creating a welcoming interface which motivates readers to pay small amounts of money. Obviously people are reluctant to pay for something they get for free. So give them something in return directly, even if most publishers are not already on board. Badges and other rewards are possible. It's simply a question of great UI, making it default for everyone, and creating a community vibe around it.
What prevents Mozilla from giving paying users some additional features like in-browser Ad-Blocking? It's as simple as forking uBO and letting it run in the background. Then advertize monthly payments with full-fledged privacy, security and browsing improvements thanks to blocking ads. (We are talking about defaults here. So after a new install every new user is greeted with something like "Pay x$ per month and get a completely ad-free web and support your favourite creators and create a new ecosystem for journalism")
Then over time when publishers are on board Mozilla could provide the interface that makes communication between both parties possible, similar to what Patreon does.
This could for example be done as an experiment in Germany first, where Firefox has 30% market share on Desktop. Reaching 1 Million active users could send a strong signal to publishers to honor paying Firefox users.