I would recommend you create a support ticket with Tableau support to troubleshoot this.
Personally if it were me though, I would probably uninstall all drivers, start clean and then install the latest drivers only. I can't say this will work, but it's what I'd try.
Support reckon it's a problem on my end that I will have to resolve with our database admin. Still doesn't explain why it worked in 9.3 but doesn't work in 10.x.
I'd push back on that then.
However, they might be right. It could be a problem with your configuration for the latest version of the driver. I feel like this is especially true given that you cannot connect through workbench.
I'd clean up your drivers and get it working through workbench then push harder on Tableau if it's still not working.
Downgrading to ODBC driver 3.51.30 and removing all others allowed me to connect to the server through Tableau Desktop's MySQL connector. We have also discovered that the user still has an old format password hash, which is why I can't connect to Workbench. I'll keep working on this as it's not an ideal solution but at least I have two ways around the issue now.