It’s also used to stifle 2A rights. Many states require a license application which intentionally asks not whether you’ve ever been convicted of a crime but whether you’ve ever been arrested, which can then be used to deny you.

Protecting the 2A doesn't have to come at the expense of the 1A. The real problem is the license criterion that presumes guilt, not the accurate reporting of the arrest.

In what ways has it been used to deny a license application? Please give specific examples.

Specifically it tends to be referred to as a test of “good moral character”. So depending on what the reviewing officer thinks about your arrest (not conviction!) record you can be denied due to lacking good moral character, which is obviously a completely arbitrary determination hence why that practice is working its way through the courts.