There are restrictions on free speech in public, I cannot yell fire into a crowd, I cannot incite a riot, I cannot tell people to vote for Jo Jorgensen while I am waiting in line to vote. Same with voting, I cannot vote twice even if my freedom of expression wants to.
This issue at hand is with those rights, they do not logically make sense to extend the spirit of the law via allowing some of those prohibited things via privileges.
This is why the 5th is such a good parallel, because I have an inherent freedom of movement but I don't have an inherent right to do it via the public roads, using an automobile. I have the privilege to do so via a licence. Now I can own an auto, I can offroad all over my property, that is my right but as soon as I hit the public pavement it becomes a privilege.
going back to the first, I can yell fire in my house all day long that is my right, but if I pick up the phone and utilize the public networks via dialing 911 and yell fire it is a very different situation. Again, it just does not make sense for the 1st to have some of it's limitations extended into privileges.
My overarching point is the 2A issue is not settled, many tend to take one stance or the other, but it is very much in flux, that will decide if the 14A issue is really a moot point.
Restricting CCWs in a manner that would absolutely not fly for free speech, voting, or some other right isn't the issue. The issue is that some states have set up discretionary license issuing schemes that in practice violate the 14th amendment.