There are many reasons why the North was so eager to invade the South, many of which was for plunder, destruction, and to punish. But, early on, the North did invade the South and gave protection to the institution of slavery to those loyal to the Union. The slaves were not outright freed in many areas of the South conquered by the North, and many slaves in areas first conquered, were not freed until the last year of the war. There are cases of runaway slaves were returned to their owners if they were loyal Unionist by the invading Northerners. The North did recognize the right to own slaves and protected the law and its institutions. West Virginia being the last slave state to be admitted to the Union was seceded by the North from Virginia is a very good example of Northern protection of slavery. Many slaves, protected in their status by the Federal government, were used as labor and kept on plantations to pick cotton for the North. This cotton was so important to the Northern military, its level of importance was just under defeating Confederate forces; providing the freedom to the people who picked and bailed it was very low on the list for a great many "liberated" slaves.
"The South split from the rest of the country to preserve the right to have slavery, and to keep Negro equality with Whites from occurring."
Part of your point is only arguable to a handful of the first seceding states. The treatment of states rights and the threat of military invasion was the biggest factor for a good percentage of the seceding states. The South split from a government taken over by a regional radical party bent on the destruction of the Southern culture, economy and political power. The secessionist South was determined to form a more perfect union for themselves to perserve their right to prosper peacefully and to put an end to the sectional strife that had been boiling over for nearly fifty years. The secesstion of these states was even applauded by many of the most radical abolitionist. How much the issue of negro equality had in the the issue of secession is very debatable since negro equality did not exist anywhere in the United States and did not until the mid-20th Century. Some freedmen or those blacks that where never slaves did already had some equality in some of the most radical secessionist states. In Louisiana free blacks could and did vote prior to the Civil War; in Mississippi free blacks owned and ran their own businesses and were very important members of their communities. Others were not treated equally, but this, again was a national problem, not just a slave state problem.
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David Upton