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".

[0]: https://share.confex.com/share/120/webprogram/Handout/Sessio... (slide 30)