Ok, prepare for something probably slightly wrong.

The Thalmor ban on Talos worship has a lot of things going for it. Some sources indicate that they think that banning his worship is the first step towards "killing" the god.

Maybe this might work in the specific instances of the different cultural interpretations of the Eight Aedra (as in you might in theory be able to "kill" Akatosh if you removed all who thought of him, but Auri-El would be fine, as would the divine body Akatosh is drawn from), but this would almost certainly not work for Talos, who amongst his powers includes things like "telling the universe to fuck off, I totally exist as a free-thinking individual". The fact that his heart is the heart of the world might help.

The Thalmor want him "dead" for a few reasons. Firstly, as he and the world share a heart if you could kill him, maybe the world would die as well. Secondly, even if that wasn't the case, TalOS appears to be holding reality together right now, though in this turning of the dream Akatosh might also be helping, so getting rid of him might destroy the world the Thalmor see as their prison.

Finally they want him "dead" / to ban his worship because it divides the humans more. They got the Redguard out by forcing a horrible peace treaty on the Empire, and if they could get Skyrim out by the ban of Talos worship? Then humanity would have returned to a bunch of nation states for them to crush.

Banning of Talos worship has nothing to do with Dragonborn or shouting, Akatosh and Kyne(areth) worship would have more to do with that.

Also dragons. They can learn the way of the voice (by which I assume you just mean shouting) without being Dragonborn. And Redguards - the Ebony Warrior is a Redguard and he can shout, though he might be a demi-god. Also probably anyone could if they really wanted to.