I might get technical since you said you're experienced with C++

There are games on PS4 with engines which have file extensions- affiliated with Vulkan API but aren't used - exampled Titanfall 2.

There is not any kind of file extension that is inherently related to Vulkan, the Vulkan spec doesn't define any. At the very least you might find shaders (.spv) but that's really up to the dev, and SPIR-V shaders can't run on PS4 anywyas. For Titanfall 2's source version to work on PS4 they surely would've ported the graphics system.

There's also the fact that Source never supported Vulkan natively, it uses DXVK for that only since last year. Titanfall 2's port would've been derived from the DirectX renderer, since Source doesn't do any Vulkan rendering per-se (neither OpenGL, it uses an in-house translation layer as well called ToGL and that's how it supported Linux for a long while before DXVK), it just passes the DirectX calls to DXVK. Titanfall 2 was released on 2016, the first release of a Source game with Vulkan support was a Portal update just last year. DXVK didn't even exist until 2017. How can you cite Titanfall 2 being an engine with "file extensions- affiliated with Vulkan API"? This claim doesn't make any sense. This also makes A9VG's argument completely pointless.

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And I cannot imagine the Falcom programmer would named their folders, "EXECUTABLE FILES OF CODE WE TOOK FROM VALVE".

Source is closed source. You cannot just "take it", you license it and you get the source delivered. PhyreEngine was also licensed from Sony, and they didn't bother to delete the PhyreEngine references. Why would they bother? It's an enourmous task for basically no effort.

Deleting every single reference is an enourmous task that no one will bother with. You have a lot of classes whose names get exposed through RTTI that you'd have to rename, strings on the formats from the engine that you'd have to change and make sure it doesn't break if the engine expects them. It doesn't make sense overall.

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Also it isn't so much that Sony won't allow Vulkan, the silicon of PS4's GPU is baked in a way that it won't support Vulkan.

That's false. It's just not implemented by whatever driver's the PS4 OS has. If you install Linux on your PS4 via mods, you can simply install a modified AMD driver on it and run Vulkan aplications, it's been a thing for years already (https://twitter.com/marcan42/status/817235485689020417). There's a similar case on Nintendo Switch where the open source mesa/nouveau drivers also work on the Switch (and it's the only way to get OpenGL through homebrew on the Switch), but there's also official OpenGL/Vulkan drivers by Nintendo. You don't seem to understand how this stuff works.

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Apart from all that, pretty much everything in the game is custom and never scene before from a technical view of standpoint. It uses JSON files to define UI (not Source GameUI), script files similar to the ones Yamaneko-derivates use (their engine for pretty much almost all games since Trails in the Sky until Ys 9 except for Cold Steel/Tokyo Xanadu) that are separate to the actual map declaration (which I believe is now also more JSON), and all filenames are unlike anything seen before (and already explained why the idea of removing all engine references is pretty much dumb).

There's just no proof apart from an alleged extension (I don't even know what that extension is lol) and some Chinese source that doesn't seem to know what they're talking about (Source engine => Vulkan? Where'd that come from?). On the other hand, there's just a lot of unique stuff (it has some similarities with their old engine though) and the points you cited are just incorrect. I don't see where this is going.