A long story! :-)

There was indeed a famous controversy about this that people commonly feel the professors (who were opposed to the oath) "won". I remember attending a conference at Berkeley that was celebrating the 50th (?) anniversary of the controversy, and the attendees were all talking about it from the perspective of the oath having been defeated. I got up and said something like "why are you all celebrating? there's still a loyalty oath and it's still being enforced!". Which didn't go over very well.

I used to know the details much better (having read several books about the controversy), but more or less as I think I recall it now:

There was an attempt to create a special oath for university employees (based on suspicion of their potentially having Communist sympathies), and the university employees rebelled against this. Part of the compromise to resolve the controversy was to have an oath for all California public employees (some form of which had already existed beforehand), and to say that people employed by UC had to take that oath, which was in some sense not singling them out as suspicious. (It might also have had wording that was somewhat less objectionable to the protestors, as there were many different formulations proposed for these oaths by different parties.)

Later, there were court cases which held that "negative" oaths were not OK (for academic employees of public institutions?) but that "positive" oaths were OK. (Like "I do not believe in Communism" vs. "I support the principles of the constitution".) So, the oaths used in state university systems were narrowed accordingly. (The "negative" forms are akin to what you refer to, which were interpreted as a kind of political litmus test, while the "positive" forms were not -- though I think they ought to be.)

Most academic workers did not find it as offensive to take a "positive" oath that other public employees were also required to take, and so this iteration of the compromise has lasted up to the present day. There were some people in the original controversy who didn't accept the compromise and left academia (I think Fred Cody, the founder of Cody's Books in Berkeley, was one!), or went to private institutions instead. However, these were apparently a distinct minority.

Nowadays a couple of people per year in the UC system (and CSU and community colleges systems) find the oath objectionable, often because they are Jehovah's Witnesses, or anarchists, or very serious about civil liberties in the abstract. (One person I heard about was the child of Japanese-Americans who had been required to take a loyalty oath on being released from the internment system in World War II, and found the whole concept extraordinarily offensive.) I've heard from many of them over the years. There are also some Protestant groups who do not believe one should take any oaths, but they've mostly been mollified by the "swear or affirm" version which has always been available in this form of the oath. Foreign citizens have also routinely been granted an exception because of concerns that the oath could jeopardize their non-U.S. citizenship (!).

Some people also worry that the oath could make signers liable to conscription (or limit their ability to claim conscientious objector status in case of a draft). There are also court cases indicating that it shouldn't be interpreted this way, but it remains a concern for a number of people.