The graphics API isn't that big a deal anymore. It's often a pretty small part of the engine codebase. Many games, especially those built on the popular engines like Unity, support both DirectX and Vulkan (and Sony's API too).
(Technically, it is also available on Nintendo Switch, but there's also an arguably better alternative there.)
If MS and Sony wanted, they could have Vulkan driver for Xbox and PlayStation tomorrow, since they are all using AMD GPUs. And no one stopped Apple from supporting Vulkan on their systems either. They don't, since they are also lock-in jerks, which is exactly the point I was making above.
That is not really an accurate statement. Graphics performance is heavily dependent on the actual hardware design of the underlying platform. You can't wave a magic API-wand and have your executable or shader be magically fast on every hardware platform.
That is really an imagined scenario. An extremely tiny minority of games are targeted for such a wide variety of platforms. Real-time high-performance code is always tightly coupled with the hardware - if you want to squeeze performance out of hardware, you want to minimize abstractions to such a level as they don't hurt your productivity. If you're targeting consoles, you can maybe target the PC platform. But there is no way you're targeting the nintendo DS or a smartphone with the same codebase without major modifications to the graphics code.
On the contrary, most games are targeted for most platforms, to increase reach and sales as result of it. Only some exclusives of console makers are not doing that, and those are clearly outliers.
> Real-time high-performance code is always tightly coupled with the hardware
That's not common at all. Unless you want to beat modern compilers with assembly code, you only will produce something worse. Sure, there are are rare cases when using hardware specific features yields extra performance. Codec developers do that for example. But for games? Not usually needed. Shaders are provided in cross platform fashion (such as SPIR-V) and that's compiled into GPU machine code by the driver. And actual game code is commonly using some high level systems language (C++, etc.) + a scripting one.
Good performance is achieved by parallelizing the engine properly, since modern hardware is increasingly multicore.