Your 3400X Roku Stick MHL is a second generation Roku that was released in 2012 and is now considered legacy equipment. The current line is eighth generation, with this year's Ultra being considered as ninth. The 3400 is no longer receiving Roku updates. The last Roku version for it was 9.1, released in April 2019, while the roll-out of 9.4 to currently-supported units is complete or nearly so.
I can't find any information yet re which Roku models HBO wrote their HBO Max to support, but I would not be surprised if they did not include the 3400.
Yes, all of that makes sense. My Roku 3400X indeed won't update beyond version 9.1.0 and my 4210X on version 9.4.0 installed HBO Max just fine. So, it appears to me that your experience & hypothesis are accurate. I'll take your word for it that the cutoff (i.e. minimum version level) is 9.3.0. Thanks for clarifying the issue for me!