The purpose of Inbox isn't to "mediate" something for the purpose of mediating it. Elgan seems to suggest that first Google decided they had to intrude with meditation, and then came up with the idea of Inbox to justify it. The reality is, mediation falls out of the architecture of email anyway.
Most day to day interpersonal communication has moved away from email and into chat apps. Your email outside of work is largely an incoming federated asynchronous dumping ground for receipts, bills, notifications, and other things. It's become unmanageable for many people, a nuisance.
Inbox is an attempt to improve the experience of email given the reality of how it's used. All email is mediated. The way RFC822 email works is, MTA's transfer messages between services, and MUAs are responsible for the presentation of that email to users. MUAs mediate mail, for example, IMAP4 SEARCH and folders are mediation mechanisms for mail. Mail clients are free to present data from MUAs in any form they want. The fact that people have largely copied the same 'From,Subject,Date' table format that's been with us for the last 30 years isn't part of any spec, it's for lack of trying.
Inbox changes nothing about the unpinnings of mail (SMTP), what it wants to do is change the user interface that presents the mail so that it has superior summarization, sorting, searching, and clustering. That requires some form of mediation unless done completely on the client side, but doing it completely on the client side imposes some performance downsides, especially on mobile.
It's not as if Gmail doesn't "mediate". It spam filters, it has priority inbox, it automatically puts stuff into tabs like "Promotions". Even providing indexing and searching of mail is a form of meditation.
Elgan has penned an opinion piece that tries to impute some reason for Inbox other than what user interface designers and product managers have been aiming for -- usability goals. The implications about wanting to find excuses to mediate and take away control are utter nonsense.