ICO - broken by design
"So I contacted the ICO at the same time, via their web page. I attached several PDFs and PNGs of snail-mail letters, but they kept saying that "I never supplied supporting information". I finally attached them to an e-mail, but I haven't had any confirmation that they received them, despite a follow-up e-mail asking them to do so."
For ICO if you never placed your case reference number in the email subject then, yes, the case officer will never receive the supporting info you emailed.
The ICO whole case handling system is broken since they upgraded/switched to a new system a few years ago (and was possibly differently broken before then).
When you open a new case via the website you should get a automated email acknowledging the new case. However *no* reference number is allocated at this stage - someone manually looks at the email Inbox for new cases and forwards them to the "Inbox" of the relevant "business area" specific team.
Once someone from that relevant "business area" team looks at the issue (which typically will not happen for 1+ months) then they will allocate a reference number. Note: you will not receive any notification when the reference number is allocated!
In many cases the first time I see a case's reference number is when I receive the "case closed" summary email many months later. What I have ended up doing is repeatedly phoning ICO approx 1-2 months after opening a new case to get the reference number.
Note: you cannot add relevant documents to a case without including the reference number in the email subject. Also note that as the website page for opening a case limits you to attaching a max of 3 (4?) documents, does *not* let you attach ZIP files (PDFs, Word docs, etc only), and has a size limit for each document then in many situations you may need to later add additional documents via email.
There's a risk however that, as you cannot add more docs without a reference number, that your case could be closed before you manage to add those documents (if the person assigned to work your case, just after the ref number is allocated, quickly decides there's not enough information to work the case!).
"I suppose I'll have to chase them, too, to find out why their workflow is broken."
Perhaps broken by design, to help limit the number of cases they have to deal with?