While the Declaration states that they (Jefferson et al.) see it as their civic duty to fight an oppressive government (a fairly Lockean view, pretty much lifted from the Second Treatise), again, it was mostly an inflammatory document, written not for the benefit of Parliament and the King, but for other nations who might consider joining the fight.
Analogically using the Declaration today doesn't really work, there aren't any of the great political thinkers of yore around, and we don't have any comparatively great thinkers (in the same sense). We're too founded on our two party system at the present to move anywhere.
Plus, today's government would refuse to accept any rebellion, regardless of what historical document they cite. Technically, politically, the Union never recognized the Confederate States of America as a sovereign nation, but a rebellion. That's the largest and longest instance of such a 'rebellion'. It's likely going to remain such.
Of course, that's due to the fact that there really wasn't any way for the Confederacy to win. They couldn't have taken and held the north. If they did, they would have assimilated the culture they were fighting against, and would be in the same situation as before they rebelled.
That actually leads me to a slight tangent, but my favorite fact about the Emancipation Proclamation is that it didn't technically free the slaves. It freed the slaves only in rebelling states. Northern slave-holding states could remain as-is.
Moving back on topic: the Declaration of Independence continually cites 'Him' and 'He' in reference to George III. George III had no power. Okay, he had a little power, but he wasn't really calling the shots any more (he never was, his reign started almost a century after the fall of the Commonwealth and slightly less after the Glorious Revolution, which cemented Great Britain as a constitutional monarchy). One could argue the use of He and Him are due to George's refusal to intervene with Parliament (something he really couldn't do).
Again, it goes back to they saw an opportunity to start something new, even though if they worked slightly harder at getting representation, they could have.