Playstation is doing pretty well with proprietary APIs.

Playstation is a stable platform, where game devs get to squeeze every little bit of performance out of. It doesn't come out every year, with new features.

It's very different than anything on a Mac or PC.

Switch also has a proprietary API called NVN, Vulkan is more like a 2nd level API.

Any game dev that wants the full hardware capabilities reaches out to NVN.

Almost no game on the Switch used Vulkan, except for a few PC ports.

You can use normal APIs on consoles through translation layers (it works the same on the switch). The translation layers aren't the best so everyone just ends up using the proprietary 'API' in the end, which translates/gets inlined down to raw commands written to the GPU's command buffer at compile time.

Vulkan is probably the least used API across any games. Most switch games don’t use it either.

Honestly, this is the biggest division between gamers who obsess over API and people who develop gamers. API divergence is really not that big of a deal as long as you can do similar things.