1. You are right about the US players' unions. Well, almost right. Baseball has antitrust immunity for some reason. It's an odd quirk in the law. But that aside, you are correct. However, that doesn't mean the collective bargaining aspect is necessary.
2. I was an antitrust attorney, and I will summarize how I see the issues presenting in this case. This is a logical progression, not a temporal one.
A. The first inquiry is: does this sports league violate antitrust law in any way? If clearly no, then nothing else is needed. It's just a joint venture (more or less).
B. If you answer yes, maybe or even probably not, it means the sports league has possible legal exposure. So the leagues find it advisable to rely on the safe harbor of the Norris-LaGuardia Act that exempts labor union contracts from antitrust scrutiny. It's not necessary, but it is sufficient.
C. If a league can't recognize a union for whatever reason, it can still fight any antitrust case brought against it.
Here's where the nature of antitrust comes into play. Essentially, antitrust laws only prescribe "unreasonable" restraints of trade. If an economic arrangement is reasonable, it is valid even if it has anti-competitive effects. In evaluating the reasonableness of a sports league's restraints, the court will naturally look to other sports leagues. If all other sports leagues have these kinds of rules, it suggests strongly that the rules are reasonable and indeed even required for a sports league to succeed. So it doesn't matter what law is applied in European football. It's the economic details that matter.
As for why universities don't want players to unionize, I think there are a lot of potentially valid reasons. First, if a public university recognizes the players as employees, they might get covered under state public employee laws. Changing the sports administration from a system enrolling students (so to speak) to enrolling students and employees would seem to have some potential to go sideways. Are the players also going to be students? Well, suppose they do something wrong. Are they subject to the Honor Code as students, or a different code of conduct as employees? What procedural protections do they get in each forum? Do they get to choose the forum?
It's not that I think these questions are necessarily difficult. In many cases, they just entail a choice. The point is, though, that a change of that magnitude can often create a lot of problem in other areas. It's hard to predict all interaction effects between a fundamental change in athletic department organization and everything else at the school. And I don't consider it surprising that universities don't want to open these cans of worms. It doesn't make that a good decision, but it seems not ridiculous to me.
1. Despite having an antitrust exemption, MLB players still have the MLBPA and there is still a CBA between the players' union and the teams.
2. While I think not having students declared as employees was a issue back in the day for universities, I'm not convinced that is the case now. Most universities hire students as employees (even beyond work-study positions) and so I think they likely know how to handle that process. And given that players are now getting paid, I think that concern has been largely removed.
My hunch is that the NCAA's inaction has more to do with its structure and the inherent lack of consensus across the member institutions as to how to proceed. (I also think it has a lot to do with different visions for college sports among different levels of schools in the NCAA.)
The ability to pay players and how to handle this issue almost certainly varies a lot depending the the position of the school. There are clear differences between Charlotte and ECU and UNC...much less adding in the Bamas and tOSUs of college football. (And those are all FBS level schools, there also has to be some understanding between the FCS, D2, and D3 schools.) But even just focused on FBS schools, I would imagine it would be hard to get the majority of these schools to agree to a common plan for athlete pay as the circumstances at the wide variety of schools are so different.
I also would guess that the idea/threat of the largest P4 (or P2 + a few ACC schools and maybe a couple B12 schools) schools breaking away from the NCAA or the current FBS structure also is throwing a wrench into figuring out how to move forward with student-athlete pay. I could see the SEC/B1G schools not favoring any formal agreements right now that could later be an obstacle into them breaking away from the current structure to create a unified nation-wide football conference for the largest schools.
I also think that a lot of university presidents know that paying players, at least at the levels we're now seeing, is unpopular among a number of their faculty. It was one thing when the football or basketball coach was making significantly more than most professors, how do they handle it when the starting QB or PG is out-earning them, as well.
For those reasons, I think a lot of university presidents - and thus, the NCAA - may be taking a "let the courts figure it out" position before the NCAA eventually takes action. One, whatever the NCAA tries to create is going to be difficult to gain agreement between themeselves, much less the players in a union with a CBA. The more the NCAA allows the legal process to occur now, the more that some basic parametera have been established that might make the eventual process easier. Two, my hunch is that the university presidents (and, therefore, the NCAA) are working a bit of a "blame it on the courts" angle with the faculties/stakeholders at their universities, whereby they can pass the buck for these changes and not be fully responsible for them.
Also, as an aside, one place where schools may have come out ahead, in a way, is in the House settlement with regard to revenue sharing. The four major leagues in the US (MLB, NBA, NFL, & NHL) each pay about half of their revenues to players via salaries (all are between 48% and 51%). The House settlement calls for the NCAA to "share" 22% of revenue with players, a much lower percentage than what the major leagues are doing via CBAs.