For 32-bit, I searched around the net for some offline installers, took them to where my computer collection resides, and started work. Turned out that every one of the downloads was faulty in some way, but at least I found out which file causes the thing to error out - it's the RDR Services Updater - probably the least important file in the entire package. But they think it's sooo important that the entire install has to be cancelled and rolled back.
So, for Windows 10 32-bit, both of them had not been able to proceed past 20180, which is quite old. I did a total uninstall, then manually deleted the left-over files, registry entries, and scheduled task. Then I got the latest 21223 from their site, and it installed fine. Just had to disable the cloud stuff as usual.
For windows 7 64-bit, most of them had 21078 which is close enough, so just locked it down so it couldn't update itself.
And lastly, Windows 7 32-bit, most of which has failed in some way months prior. If they had version 21xxx I just locked it down as above. For those with version 20xxx, it was uninstalled, same as I did for the windows 10 machines, but I found an offline installer on one of the machines, version 21043, which actually WORKS - so this was used followed by the usual locking down.
So, this gets all the Windows 7 machines to a fairly recent, working version of Reader, which will never get updated.
There's a couple of downsides of course, if you sign into adobe via the reader you can't do that any more, and most noticeably whenever you open the reader it complains twice that the updater can't be started. A small price to pay.
Some things to note: If you decide you don't want Adobe any more and uninstall it, it leaves behind the updater and a scheduled task to run it, which it does. I'm not sure what they are hoping to accomplish by doing that. You need to remove all the remnants yourself. And they leave behind all the registry entries (except for the policy which gets removed), so in the interest of cleanliness you may as well delete all that as well.