Turns out, "Blocked URL list" under "URL filtering" under "Content Filtering" had '*' for the entire network (not just this specific SSID, not any specific VLAN or device group). Not just on that network, either.
Removing it resolved the problem.
I'll have to dig around a little more to see why all the other SSIDs (4+) and wired devices across several VLANs worked, or why that particular SSID was working up until a few weeks ago, yet the root cause seems to be a bad case of misconfiguration with multiple rules and policies in different places stepping on each other where which rule takes precedence is unclear.
(While troubleshooting this, I was wishing for something like Active Directory's "resultant set of policy" tool that gives me a prioritized list of rules and policies applied to a specific entity or a set of them. Makes it much easier to troubleshoot issues, especially when multiple rules with overlapping scopes and restrictions are scattered across OUs. (In Meraki's case, it's firewall, group policies, SSID-specific policies and rules, content filtering, port configuration - which can all step on each other and where the result can be quite unclear - like in our case.) That, and better log aggregation and querying. Is this too much to wish for?)