A comparison between the 1776 rebellion and the 1860 rebellion. RW = the Revolutionary War, ACW = the American Civil War.
The rebelling party was a full member of the body politic:
RW: no. ACW: yes
The rebelling party had willfully and freely entered into the government from which it was rebelling:
RW: no. ACW: yes
The rebelling party had access to full representation on the national stage:
RW: no. ACW: yes
The rebelling party had attempted to have their grievances redressed, and hostilities began before they declared separation and independence:
RW: yes. ACW: No
The rebelling party began their rebellion after losing a free and fair election in which they were a full participant:
RW: no. ACW: yes
The rebelling party made clear in their documents of separation that their main concern was protecting chattel slavery of the African race:
RW: no. ACW: yes
The rebelling party made clear their right to separation through war and de facto independence:
RW: yes. ACW: no
How are these conflicts remotely similar?
Try a different analogy, namely the modernn EU and Brexit: (B = Brexit)
Was the leaving party a full member of the body politic
ACW yes, B yes
Did the leaving party enter willfully and freely into the government
ACW yes B yes
Did the leaving party have full representation in that government
ACW yes. B yes
Did the leaving party attempt to have their grievances redressed (Leaving our the beginning of hostilities here since it’s not applicable to Brexit)
ACW yes*. B yes
The leaving party left because of substantial disagreement with the government over what they believed to be a critical issue
ACW yes (slavery). B yes (immigration policy)
* I take issue with your judgement that the South did not try to have their grievances redressed before resorting to war. Most of the significant historical events of the first half of the 19th century dealt with the South trying to secure its interests through the political process and the compromises that were reached with the North that postponed hostilities. The war would have likely begun at least 40 years sooner had the South truly not attempted to have their grievances peacefully redressed.
I also am not addressing the legitimacy of the reason for secession in these two cases, only the legitimacy of the act of secession. The act of secession itself can be legitimate even if the reason for that secession is repugnant (as was the case for the Confederacy). There are those who might argue the same for Brexit as well, claiming it’s based on xenophobia and white supremacy. I reject that characterization, however. A sovereign state has the right to control it’s borders and immigration policies.