Head of History here. I teach KS3, GCSE and A Level. I'm also an examining marker. I don't tutor, I haven't got the time. Will try and be as neutral as possible.
Firstly, a uni student isn't really a tutor. They don't have any background in pedagogy (how people learn) and their subject knowledge will be juvenile compared to an actual graduate, with a Masters, a PGCE and several years of teaching experience. You get what you pay for.
Teachers advised against history but DS did it anyway. OK, that's a done deal. We are where we are. It's actually OK for a kid to want to take a subject because they have a genuine interest in it.
He wants to do it at A Level - you really need to listen to the teachers this time round. If DS doesn't have the academic ability to do it, then that can't really be changed.
The first thing I do in a revision session is work out which areas the kids are weakest in. The way I do this is by getting them to RAG rate their own knowledge. I don't know, they might be amber on causes of the Cold War, green on increasing tensions and red on detente. This tells me where my highest leverage points are. If they're red on everything, then we've got a big problem.
I'm unsure how you'd get the Cold War confused with something else. DS will be doing 4 different strands. He really does need to know the very simple key words that go with each, e.g. Cold War, Weimar, Empire, etc. These aren't really very confusing.
This is alarming: 'Subject knowledge is a key area of weakness, and DS apparently said to him that he doesn't make notes in the lessons because his teacher isn't good.' Firstly, making notes is the way a learner transfers information from their short term memory into their long term memory, i.e. the way they learn. Revision methods such as quizzes, flashcards and mind maps are designed to help students recall key facts from their long term memory. But if there aren't any facts in their long term memory because they didn't do the learning, then there's nothing for them to recall.
In terms of you paying for a service, depends, if you're paying for a revision tutor, sure, you're correct. However, from the sounds of what you've said, your DS seems to need an additional history teacher, so a revision tutor won't be able to help him.
Secondly, not sure about your DS's school, but our GSCE students are assessed every two weeks to check their understanding. What were his grades like for the whole of Y10?
Anyway, if I were you I'd sack off the revision tutor. It will be almost impossible for him to re-teach your DS an entire curriculum in a few months.
I'd also buy the revision guides from the exam board - they'll be published separately for each unit. They're not too overwhelming, i.e. quite small. This will also give your DS the flexibility he seems to need to complete his revision. But you'll have to do it with him. I should imagine the school also provide their own revision resources. You could ask your DS's teachers to give you the course spec. He might not do too well in his mocks, however, following the revision guides could bring him up to speed for his actual GCSEs
Have a really good hunt around for other resources, for example, on Netflix at the moment there's a good 'Turning Point' docu about the Cold War. BBC Bitesize / iPlayer also has some great stuff. You can set this as 'research' for your DS and then watch together. Get him to explain why this is relevant to his GCSE. This shouldn't be too much of a ball ache if your DS has a genuine interest in history.
Also check out 'Save my Exams' and Seneca (where there are quizzes). I don't know which course / units your DS is studying, but 'The History Teacher' on youtube might cover some of the content in her student friendly revision series.