It wasn't about the existence of the monarchy, it was about the role of the Constitutional Monarch in Great Britain. Under the Constitution at the time, the Monarch had 3 defined rights: the right to be consulted, to encourage, and to warn. Beyond that, he was a figurehead with no power to make anyone in government do anything or accept any of his decisions. He was expected to do as Parliament directed him to do. The trouble was, the only absolute prohibition outlined in British law about the marriage of a monarch was that he/she could not marry a Roman Catholic. Wallis Simpson not being RC, Edward thought he was on solid ground to declare that he was going to marry her. The PM and the Cabinet, however, had been advised that Wallis had ties to the Nazis, and they believed that she was possibly a spy; they felt that allowing her to marry the King would also allow her to get access to a potentially large number of government secrets, which they considered an unacceptable risk.
The British Constitution, like the US Constitution, has a number of what we call "penumbral" implications, and one of the most widely accepted was that as a purely titular Head of State, that it was expected that if the Govenment and the Monarch had a clash of wills over any issue that was seen as having the potential to damage the nation, that the Monarch would yield to the will of Parliament on that issue. As the Cabinet were convinced that Mrs. Simpson's marriage to the King would be injurious to the Nation, the Government demanded he yield or get out, never imagining that he actually would choose to go. (It should be noted that Edward also did not think they would actually make him go.) So, the real issue was whether or not the Constitution actually gave the Parliamentary government the power to make the King obey them in matters that did not directly relate to the operation of the Government.
(This is a very simplistic view of the situation; it was, of course, more complex than can be explained in 2 paragraphs. Still, that was the root of it.)