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Lots of confusion still about the very concept of an intermediary (see the many comments below), so time to sort this out.

First, an intermediary is traditionally a provider of a service to facilitate a transaction between original producer and ultimate consumer.
Without the intermediary, this transaction would either be very costly or impossible for the end participants to undertake. Bundling less-than-truckload wares into a shipping container is such an intermediary service. Wholesalers are intermediaries.
Indeed marketplaces are intermediaries when they are not operated by the end participants. This has led to quite a bit of confusion among economists who are used to think of a market as an inter-entity abstraction and not as an entity in itself.
The word "intermediary" has acquired such a negative reputation that it bears pointing out that intermediaries almost always start out by providing a valuable service: bundling, aggregation, risk bearing, vetting, etc.
Our unease with intermediaries — middlemen — comes from the fact that they hold a strategic position within the transaction which they can exploit as soon as they establish a certain dependency for market participants; vulgo: (market) power.
This triggers the fundamental transformation from multiple adversarial optimization problems (the Edgeworth box) to a single problem: Maximize intermediary profit subject to all other participants getting scraps.
Fast forward into the information age. Instead of bundling packages into a container, the intermediary facilitates the transaction of information goods.
This comes in the form of aggregating, bundling, forwarding, and generally transforming information packages from the suppliers for the consumption of the end users.

It also includes, specifically as a service, strategic (partial) disclosure.
The archetypal information intermediary is a notary service. The notary confirms the veracity of a certain claim but withholds all the ancillary information required to make such a confirmation.

This is why we speak of "trusted intermediaries" in the information age.
This is important to sort out the confusion about whether it matters if there is only one intermediary (a gatekeeper) or if the market for intermediary service is competitive.
Two distinct qualities about information goods: information cannot be unseen, and it cannot be withdrawn. So a notary, a "trusted" holder of an information good, cannot credibly demonstrate that they destroyed the information collected on behalf of their clients.
So there is already a fundamental transformation by choosing a particular notary: once the notary is in possession of the ancillary information, this private information cannot be withdrawn by the client. The notary now holds a hostage.
A third characteristic of information goods is that data typically doesn't scale by aggregating like data, but by collecting and interacting diverse data. This value creation is completely opaque to the outsider.
Now. The fundamental transformation of blockchains, the "Satoshi moment", is to rethink this information supply chain by disclosing most of the data we previously considered in need of protection, and holding the identities of the transacting parties behind signatures.
This is done in order to limit the opportunities for holdup: the ability for intermediaries to extract rents from (temporary) information asymmetry advantages.
So miners as producers at scale become information intermediaries as soon a they make it impractical (prohibitively costly) for peers to facilitate a transaction directly. The goal is to limit their power from that intermediary position to also become gatekeepers.
The technical boundary for this is widely known: It's the Byzantine condition that no single entity can (publicly or secretly) control more than hold more than 50% of hashpower.
The more subtle boundary comes from the technical characteristics that determine the market structure of hash mining which lead to location advantages, scale economies and learning curve effects, barriers to entry, and ultimately market concentration; vulgo: power.
The gatekeeping in this case doesn't come from the power of rejecting a proposed transaction outright (which requires monopoly control), but from the power to control the order of transactions (which hinges on the ability to monopolize the next few blocks).
The less sinister effect is that a powerful miner can delay single transactions, and the more sinister effect that the miner can change the precedence order of the transactions wholesale (via "selfish mining").
So in summary, miners become intermediaries as soon as peers can no longer vertically integrate into hash production, which happens quite early. They also become gatekeepers as soon as they can defend a certain precedence order of transactions against other miners.
This thread was brought to you by the #BeigeAcademy of Old-School Industrial Organization Explains Everything.

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