What's that purpose then for a secret instruction set?

Probably so they can change it at will without having to worry about breaking other people’s code.

Companies either want to be earning money from an API, or they want to keep it secret.

An internal API which people reverse engineer and use is just going to lead to hassles later when you want to change the API, when people start writing bots or abuse the API in ways you hadn't imagined, or expose bugs in the API the official client didn't.

I haven’t heard it about CPUs, but worries about patents also could be part of the reason. That is/used to be a fairly common argument as to why GPU makers do not release hardware specs.

Is that because they know they are violating other people's patents and don't want to be caught or just that the patent system is such a nightmare that even though they didn't intentionally do it something they've done is probably infringing on another company's patents?

Yes.

Vendor lock-in for their IDE/toolchain. If the ISA isn't open, you're going to be paying the vendor for a license to compile new code.