As far as I know, there is no viable solution to pay $0.01.
Which is a pitty, because $0.01 per page would be a CPM of $10 which is crazy high. For most publishers, that would be a big increase in their earnings.
The concept of a tip jar that you'd fund and then be able to disburse on participating websites.
Then payments could just be aggregated and paid out periodically from the tip jar provider to the publisher..
But you need some kind of critical mass for this (both users and publishers), and I don't think anyone has cracked this yet..
If you make a few app store transactions in a day, I believe apple aggregates these into a single transaction that's posted to the card. That model could help here, though obviously it won't help for ultra low usage though. Pre-paid models might work, but those have their own challenges, and all of this points towards a centralised gatekeeper knowing everything a user reads (or at least which top level publishers they read content from).
Also, a lot of media sites use the personal information they collect with your account for tracking, which they'd probably lose some of since they won't be the one billing you anymore.
There are micropayments that do sort of work - for example, cellphone plans that bill per megabyte of "data". But even then that only works because mobile operating systems and apps do not provide a way to anticipate data costs. There is no cost displayed next to each link you tap or YouTube video you watch. If there was, nobody would use their phones. Usage-based billing is actually one of my pet peeves for exactly this reason.
What we really want is an all-you-can-eat model that just gives us what we had back in the Advertising Will Pay For It era of news. Except that will never exist because everyone knows how platform capitalism works. Once you condition users to pay a fixed amount per month, then all the creators on your platform - or, in this case, news organizations - are playing a zero-sum game. Because they are being paid out of one pot and the only way to make that pot bigger is to do the platform's advertising job for you. This is what happened with Spotify and Netflix, with the added intrigue of the platform playing artists off one another with original content and exclusives.
On a more technical level, transaction networks are expected to provide fraud insurance, and that bills at a tune of 30 cents per transaction. Any mechanism to shave pennies off your wallet will be abused to commit untold amounts of tiny fraudulent transactions. For example, imagine installing a browser extension that, in the background, starts loading up your news articles without the user's knowledge. That's the sort of thing that will kill a microtransaction system long-term.
There are a few coins in the crypto space that could legitimately handle this, such as Nano. Its transactions are fee-less and generally confirmed in under a second.