My H9 Control app constantly offers to download it, which I've tried many times, but to no avail.  It makes sense that H9 Max users are supposed to simply update their software to get new algorithms, so why does the control app offer it?  Should I update the control app?  How?

Yes, if you haven't installed the latest version of H9 Control, you need to do so.  You can find the latest installers here:

https://www.eventideaudio.com/support/product/663/installers

You can check what version of H9 Control you have by clicking on the Settings button in the app's bottom menubar and going to About.  Version 2.8.1 is the latest version.