To clarify, zIIP processing is only relevant under z/OS. Linux processing runs under IFLs. But that's not to say you can't get some offloading in Linux. By design, mainframes can hardware accelerate some processing. The crytpocards [0] are an example of this.
But you are right. A lot of shops go through a lot of effort to maintain their zIIP window in their z/OS processing. And a big selling point for many mainframe software vendors is that their product's processing is "zIIP-enabled".
Yes, you are right there, and thank you for clarifying that. I should have been more clear that the zIIP is useful to Linux through accessing data via z/VM. You can't offload all of the I/O, but you can offload some of it if accessing the mainframe for data. Or so I remember at this point. It's been a few years since leaving my mainframe systems programmer position.
But you are right. A lot of shops go through a lot of effort to maintain their zIIP window in their z/OS processing. And a big selling point for many mainframe software vendors is that their product's processing is "zIIP-enabled".
[0]: https://share.confex.com/share/120/webprogram/Handout/Sessio... (slide 30)