I did do a Get- after doing the Set- and it showed what I expected. But seeing the $global:DefaultVIServer.ExtensionData.Client.ServiceTimeout still set to 300 is what tipped me off. I then did a Disconnect-VIServer for my existing session I'd been working in and then when I reconnected, this value was updated to -1 and Get-Log did not time out.
So to those that may read this in the future: it appears that Set-PowerCLIConfiguration -WebOperationTimeout does not affect existing PowerCLI vCenter/ESXi connections, only future ones.
Thanks Luc for the hint in the right direction.