If you're comfortable coding to something as low-level as Vulkan, it shouldn't be hard to use whatever API the PS4 or XBox has. You'll want to customize for the hardware regardless, since it's a bit underpowered compared to PC, so the only difference will be your API functions don't begin with vk_

Comfortable doesn't mean you always have time and resources to do it for N target APIs which are completely incompatible. Game studios clearly benefit from avoiding reinventing the wheel and duplicating their work. That's the whole point of lock-in proponents. They want to tax developers, to make cross platform development more expensive, forcing at least part of them to limit their release targets. That's a dirty anti-competitive tactic.