The reason it's complicated is because the SCOTUS has ruled a bunch of different ways when it comes to what is protected speech and what isn't as well as it being acceptable to restrict some speech of students while they are at school.
Tinker is the case most people point to for students having protected political speech but it's not a blanket protection. In the Tinker case the student's speech didn't disrupt the school which was a key fact among others.
Morse determined that a students First Amendment right wasn't violated when he was punished for holding up a sign that said bong hits for jesus during a school event where they were watching the Olympic torch go by. In Morse one of the keys was that it wasn't political speech and the speech reasonably could be seen to violate the schools rules against promoting drug use. Not all of the justices agreed with that and two of the justices that agreed the students rights weren't violated were concerned the decision could be extended to political speech.
Public schools can't blanket enforce their rules all the time everywhere but if there's an established nexus between the student and action and the school they can. The ACLU points to a case (I don't have it in front of me) where cyber bullying was punished by a school due to a nexus created by it being student on student activity even though it didn't occur on school grounds.
There are rulings also cited by the ACLU about profanity and vulgarity where the school is allowed to restrict it because the student performing the action has a captive audience. In the case they cite it was during an assembly.
So basically the argument is that the walk out is not a disruptive form of political speech, this student was not in school and was given a tardy as proof, the statement was made during a private call between the student and his representatives staff and that it was not reasonably to believe was overheard so there's no nexus and there's some more about it being retribution and excessively harsh punishment. They also try to tie in some other actions where the student's possibly political speech during his debate activities were also met with reprimands by the school though not punishment. The ACLU articles say the student was not on school grounds however their letter doesn't actually say that.
The school district doesn't address the walk out as being disruptive or not, all students were given a tardy for participating in it. The school claims it did happen during school (10:00am to 10:17am) and on school grounds and the student doesn't appear to be disputing that. From my personal knowledge of the situation the students met in front of the school on the campus (I live here, my wife works for the school district as a teacher though not there). The school claims the student didn't have a quiet conversation with the staffer but in fact shouted at them and used demeaning and vulgar language and that this incited other students to shout and use vulgar language as well. The district also claims the student in his call told the staffer that he was a representative of the high schools student body. Given he had recently won a position on the student council it's possible that's the capacity he claimed he was calling in.
Keep in mind there's probably a recording of the call and having called Amodei's office I can assure you that the student would have been notified of that. So the ACLU has to get a court to agree that simply walking out of class is enough to not be in school and I don't think they'll win that point. If the districts version is accurate and the student shouted and incited other students to respond in kind that could also hurts the students case and establish a nexus between him and other students beyond just performing the action on campus. Also if he called and claimed to represent the school or the student body that probably doesn't help him either.
I'm not a lawyer and I'm just going off of what I understand of some relevant case law and the facts as they are being presented in this case. I don't support the students suspension but the version of events he's told and version of the events told by the district aren't lining up and this is one of those cases that if it were to be pursued I think would end up before the SCOTUS. Clarence Thomas in his opinion on the Morse case stated he didn't think the First Amendment was intended to extend to students in school at all. Another poster in the threads on this topic has pointed out repeatedly that the SCOTUS has basically ruled both ways on cases like this and you can't reasonably expect them to rule one way or the other based on that alone. So, in my non expert opinion it's not a clear case in favor of the student.