How can I make tags in tagsets visible to users who want to tag contacts
By design, tagsets are an auto complete drop down.
However, there is one thing we could do to improve the usability: make the list open immediately and show the complete list of tags.
I don't really see a downside to that change so I'll make a PR and see if anyone objects.

https://github.com/civicrm/civicrm-core/pull/12248

UPDATE: The feature has been accepted and will be released with CiviCRM 5.3.0.