Yes it is important to discover whether there is something on the site preventing the tabs from working at all or whether making the change to enable custom css caused the issue. If the tabs do not work with the standard styles, you need to get it working with that first.
It looks like the tabby.js script is being included on the page but not being applied, so it is possible that a script in your theme or another plugin is preventing the script from running.
To troubleshoot this you need to investigate whether that is the case, and if so identify which component causes this. To do this you could temporarily change to one of the default themes (eg twentyseventeen) and deactivate all other plugins, then if you see the tabs, change the the theme back and reactivate the plugins until the tabs break again.
Alternatively you could try using the “Health Check” plugin for troubleshooting: https://www.ads-software.com/plugins/health-check/ On the troubleshooting tab, you can click the button to disable all plugins and change the theme for you, while you’re still logged in, without affecting normal visitors to your site.