I actually agree with almost everything you say.
It would be odd were the strength of feeling to arise only because of exam fees. I can't speak for others, but for me this is just one limited example of what I think is a flawed and really damaging wider attitude and approach.
I sought to suggest that GCSEs are practically compulsory. I was meaning to acknowledge that there are not strictly compulsory. I also readily acknowledge that there are some students for whom GCSEs will not be a practical option. But for the very significant majority, they are essential and, therefore, practically compulsory, in the sense that for that cohort, most jobs will require them, as would almost all routes for further study, etc. If you asked most year 11s whether GSCEs were optional, I should imagine that the vast majority would suggest that - as a matter of practical reality - they are not.
That does, I think, make it easier to justify state funding of exam fees for all than, say, lunches or uniform. In practice, exam fees are funded by the state for most children, and lunches and uniform are not (though there are, of course, exceptions).
I can see why it seems odd that there is strong feeling about paying for exam fees. As you say, it is a relatively small amount against the total financial commitment.
My real issue (and I can't speak for others) is that the 'system' in place is rigidly binary. You either accept whatever state options are available to you, or pay everything yourself. I'm yet to hear anyone attempt to justify why there needs to be such a binary system rather than, say, the ability to 'accept' / receive the 'standard' funding for tuition and exam fees, and then put this towards a more expensive option. The state would only be paying that which each child (or their parents) has the right to require the state to pay by choosing the state school system.
Exam fees are a drop in the ocean. But when you add this to the existing inability to receive any state funding for education because of the choice to look for an option other than that which happens to be the available state option, and then the addition of VAT on those fees, it feels like a yet further kick in the teeth and, importantly, one for which there is no principled justification.
I feel more strongly than I ought to because I fear that the current approach to private education will dismantle a number of valuable schools, some of which have been built up over decades. I appreciate many will see that as a good thing; I'm afraid I don't.