And all the examples of words with a privative a before them mean the negation of, or the absence of, the following word.
From a purely etymological perspective, you are correct. However, the word "atheism" is not necessarily perceived that way by every speaker (especially those who tend to spell it "athiest"), and we know that modern English only has a productive prefix
a- that attaches to adjectives. So you are committing an etymological fallacy to argue the meaning of this word based on etymology. I'm sorry, but that is a recognized fallacy.
I'm simply not going to consciously use words to mean something that isn't in their structure. Adding a privative a before the word theist means I'll use it to mean the absence of or the negation of, theism.
But it doesn't work with other nouns. So you cannot call someone who is not a politician an "apolitician". The productive privative prefix in the case of nouns would usually be "non-", as in "non-politician". And you can have a "non-theist", which is simply anyone who is not a theist. The word "atheist" is an older word in the language that came into English via French with its Greek morphology already intact.
Non-believer is too broad. It is not a synonym, because "believer" has a far more wide-ranging scope than "theist" does, even though theist is already rather wide.
It isn't too broad if you use "believer" as a synonym for "theist", which we frequently do in this forum.
Non-theist might work, but it's a synonym of atheist. You propose that we should have the words "non-theist" and "atheist" mean different things, even though they are functionally equivalent?
When you really examine word meanings, it turns out that there are no pure synonyms. However, "non-theist" can be a euphemism for "atheist", so it is used as a synonym sometimes. Normally, it just means anyone who is not a theist, which could include people who have no concept of a "god".
If you define two words, with one meaning, roughly, "can believer", and the other being a word with a privative before it meaning "not a can believer", then I would be a "not a can believer".
We are no longer talking about the prefix "a-" here, so I'm not getting your point.
If I worship a goddess named Isabel that you've never, ever heard of, would it be right to say that you believe in my deity, or that you lack belief in it? Of course you lack belief in it, because you've never even heard of her.
Well, the question here is one of how people use the words "atheism" and "atheist". I would maintain that most English speakers associate the concept with rejection of belief, not mere lack of belief. So, if Tarzan doesn't know about gods because he was raised by animals in the jungle, one might refer to him as a "non-theist" more readily than an "atheist", which carries a lot more semantic baggage.