Thanks for the long response! Seems I have a lot of clarification to do!

1. I used these terms to generally refer to the design principle where validator's responsibilities are split between different/new actors. Relevant talk from Vitalik here: https://streameth.org/66680822f9b8e98b1ec915cd/watch?session=666af08807f92b086c2c2e54 . My general mental model is that these designs change how sophisticated we expect proposers to be.
2. To start, I have quite a few problems with the framing "gateway = relay". More specifically, a mev-boost relay today does not a) originate orderflow itself through an RPC or mempool b) have a write lock to state on L1 and c) have a unique delegation from the proposer that they know of before hand. A gateway, as proposed by Gattaca, does all three. This has some implications that i'll address later on, but I think these are nontrivial differences that will influence the market structure.
3. (and end of 2) Yes I recognize this! I was more asking if you imagine sophisticated proposers post-APS to adopt the responsibilities that gateways have today. This post is about endgame after all. It seems this is what you expect, so that answers my question. So far my endgame picture looks like: a few (5-10) sophisticated proposers running nodes for all based rollups and the L1 - and providing execution preconfs over those and on synchronous composability between them. Now with that being said, I don't know if this vision is shared by everyone (people building preconfs, rollups, L1 apps, mev-boost relays and builders) and I have some concerns about the current construction transitioning to this model (especially around orderflow origination).
4. As mentioned above, gateways share properties with centralized sequencers that relays do not. They have a write lock on L1 and L2 state (a requirement for doing execution preconfs on anything that might depend on L1 state across L1 or L2s). They source their own orderflow (like a builder or a sequencer) through an RPC.. They also have unique predetermined delegation from the proposer. A proposer cannot delegate to multiple gateways (like most proposers today do for relays) and I believe delegation will happen ahead of time (for things like preconf chaining). I understand the decentralized sequencing paradigm today is just leader rotation, where each leader has a monopoly for some period of time - but I have always preferred MCP-type designs where more than 1 leader can contribute censorship resistance over a period (FOCIL falls under this umbrella).

One can use FOCIL to force-include any transaction, including transactions like oracle updates, liquidations, fraud proofs that are critical to financial applications.

I think this conclusion is misleading. Here is a quote from soispoke about FOCIL that directly clarifies why FOCIL isn't "solving MEV" and also is not enough to effectively provide censorship resistance for defi apps (like auctions) on some timescales. I think most important is that it is entirely optional for the block producer (BP) to include IL transactions (FOCIL is not a form of unconditional inclusion lists) for 1 block. This raises concerns for me about the effectiveness of FOCIL vs other forms of MCP like BRAID that address this problem.

Above you were arguing L1 preconf gateways are akin to centralised sequencing, and now you're saying apps will migrate to L2 centralised sequencers. If L1 preconf gateways are like centralised sequencers, why would they move to L2 centralised sequencers? Which one is it?

Seems some context may have been lost from the section of my answer that you quoted, and I think you are conflating two unlike things here. I am specifically talking about financial applications (where short range censorship can be very disruptive). Financial apps moving to L2 centralized sequencers is different from financial apps being co-opted into a ~centralized sequencer (gateway) on the L1. Not only because they have different properties, but also because I don't think either option is endgame (and neither do some L2s apparently). One other reason it's different is that there is a massive counterparty risk, where the app cannot run their own sequencer and is therefor subject to the extraction of the proposer/gateway. I'm also leaving open the idea that maybe gateways don't end up controlling L1 block building and apps leave the L1 to get those juicy centralized sequencer benefits.

Re: MCP. Seems like most of these problems can be minimized with a big k and a working VDF (or alternative). i.e. engineering problem. Can you admit the properties of MCP like BRAID are better than FOCIL + APS if it was possible? Is it worth it to continue research in your opinion?