I experienced a related situation when I was supervising a written exam as TA: I caught a student cheating who actually (though probably accidentally) admitted cheating ("I couldn't read anything" - yea but already trying to read other's answers is cheating).
When telling my prof, he decided to nevertheless have the exam graded regularly. His explanation:
In a legal sense, we'd be in a situation where my word would stand against the student's word - we did not have hard proof. Our claim of the student cheating would therefore be very weak.
This is something I consider extremely important nowadays: given the difference in power between student and examiner and that even confessions of cheating are not reliable indication of actual cheating*, it is IMHO extremely important to accuse students on the basis of solid evidence only.
In his experience, students who cheated/were caught cheating had always anyways failed the exam due to lack of knowledge - which was actually the case.
- The official penalty for cheating was failing the exam, so correcting the exam and failing the student on that basis was safer and just had the same outcome in terms of the need to redo the exam.
- In terms of pedagogy, they had had the reprimand I gave them during the exam plus everyone else had seen someone being caught cheating.
* While marking each homework that is considered cheating 0 without any interaction with the student does not induce any bargaining problem, I gather that it is common to take a more strict view on students who do not admit cheating when accused. I'd argue from that that students may expect to minimize their losses by confession - which would be sufficient to put most interviews with students who are accused of cheating into the realm of that study.
I found a veritable source for every single one.
Your situation is similar to my catching the cheating student: there's a definitive smell, but not a solid proof: that would require you finding evidence that each student did actually access that web page (during the time of that homework) and maybe even proof of them copying the relevant part.
Without such proof, the professor may still deliver a lecture about cheating or depending on what is up to their discretion assign some more homework that is not found online, announce that homework grades do not enter the final grade, ...
As such proof is not realistically possible, the usual way out is to allow all kinds of sources but require proper citation. This is not only a good excercise of how academic writing works, but it also reverses the burden of proof: regardless of whether the thought was the student's own, it's up to them to check whether the thought has been published before and if so, cite it.
Thus, a lack of citation is far easier to proove than a student using forbidden study material.
Is this just academia?
Something went wrong here quite obviously. I'd say: on both sides, students cheating and the prof having dysfunctional rules (whatever is done: either condemns without proper proof, or doesn't enforce rules).
OTOH, academia is like the rest of the world.
- There will be inexperienced profs (who may learn and do better in future)
- There will not only be brilliant teachers but also mediocre ones, and finally
- Students are usually comparatively inexperienced due to their age and
- may actually cheat.
I may add some context: for us,
homework itself was rarely graded, it was mostly considered an offer for self-study.
Homework that was graded were either
- reports on certain questions and I think the topics where handed out according to what was of interest for the teacher that year, i.e. new topics.
- reports on experiments - here the whole labwork performance was graded
- reports that were presented during a seminar, so the student was questioned on the topic.
And they all used the cite-properly strategy explained above.
Graded excercises (undergrad/introductory) had the excercise being done during excercise lessons with TAs being around who had an eye on cheating besides helping. Some of these excercises were supposed to be group work, btw.
- In general, noone cared how we acquired knowledge, the emphasis was on having the knowledge when examined. Failing exams, particularly during the first semesters, was very common (we had exams with pass rates at first try of ≈ 25 %).
- Collecting exam questions and studying with such collections was usually encouraged.
- Important exams were oral.