Trying to drag slavery into the document distracts and waters down it's central focus, and that is why the other members of the committee struck all that anti-slavery language out of the document.
At the time, all 13 colonies were slave states, and i'm confident the document would never have been signed if that anti-slavery language had been left in it.
The civil authority behind the declaration were the States themselves, not Thomas Jefferson, who was only part of the Virginia delegation.
You are conflating a multitude of reasons for the top reason and concluding that the other multitude from #2 down must not then be relevant anymore. Try to remember that the "solitary" Declaration is a container for 27 grievances.
It doesn't matter if slavery was #4 on the list or #24 on the list, the fact remains that it was there.
And as for "dragging" slavery "into the document". Not only was that done by Jefferson himself in the original draft, but the British Empire responded in kind. They too recognized that the colonists were upset over it because the colonists spent years lobbying the king to stop the trade.
The British government in 1776 funded a response to the Declaration(which I recently wrote about), titled An Answer to the Declaration of the American Congress.
The New York Times uses the British Empire's propaganda in order to smear the US (1619 Project)
Even the king and his men recognized that slavery was somewhere on that list of reasons for the separation.