I give you an onion
For most of her last year at Chestnut Lane C of E, Hermione was essentially marking time, while her real education happened elsewhere. The visit from Professor McGonagall had been most enlightening. Hermione, of course, had never doubted that magic was real, but her parents had (with hindsight) only ever been humouring her, and she'd ended up having to reassure them that she knew perfectly well that her 'imaginary friends' were only imaginary, after they'd sent her to a psychologist the previous year. Her demon friends were worried all kinds of terrible things might happen to her if she was thought to be 'mad', and even if places like Bedlam were things of the past, Hermione was still looking forward to being able to go to libraries and museums and things on her own without Emily's supervision once she was old enough - Emily got bored so quickly - and in order for that to happen, her parents had to think she was responsible, mature, and sound of mind. So she had lied, and when she saw the beginnings of concern on Professor McGonagall's face at the mention of summoning, she lied again. She thought it had worked.
Hermione's parents were initially under the impression that witchcraft was something in the nature of a party trick, an accomplishment which, while impressive in its own way, was distinctly less important than getting a proper education with real qualifications that equipped one for a job. They had been ready to insist that Hermione keep up her maths and science during the first year, and that if her marks were unsatisfactory at the end of that year, they would remove her from school. And the professor would have been quite content to leave it at that, only Hermione kept asking questions. After all, one couldn't expect an entirely separate society to function exactly how one's own did, and assumptions were dangerous. Look at all the misapprehensions her demons had about modern society - modern muggle society, as she supposed she should call it now - due to only having visited it infrequently since the treaty, and that was with the way some of them did have up-to-date knowledge about some things due to the nature of their powers. Forneus, for example, spoke quite normal modern French and knew all the appropriate vocabulary words for things most demons didn't even have context for, since 'knowing all the languages of the world' was such an integral part of her being; Stolas with his 'superlative knowledge of astronomy' was aware of both Neptune and Pluto, despite not having visited the mortal plane since before either planet was named;* and Buer's knowledge of moral philosophy included plenty of thinkers he had never met, and extended to the present day. But the individual demons often didn't know they knew such things until the matter came up; and they had no system whatsoever to organise sharing knowledge among themselves, even when it was very clear that not knowing such things would make it much harder for them to strategise and plan effectively.
Hermione's parents were quite interested to learn that the magical world did have exams very similar to GCSEs and A-levels, called OWLs and NEWTs; that some magical careers had direct equivalents in the world they knew, like teaching, medicine, law enforcement, civil service, as well as less glamorous professions like bartending and bus-driving, that good marks got one access to better jobs, and that Hermione would not have received a Hogwarts letter were she incapable of attaining said qualifications in terms of magical power, though how good her marks were would largely depend on how clever she was and how hard she worked. The stern-seeming woman actually smiled a little and said she had the impression that that, at least, would not be a problem, and the Drs Granger were as flattered as she had doubtless intended them to be.
They were, however, less than pleased to learn that, by law, they didn't have the option of taking Hermione out of Hogwarts after her first year, certainly not without the consent of her teachers. As a muggle-born, once Hermione was enrolled in Hogwarts, she was contractually obliged to stay there at least until passing her OWLs; if Hermione's parents chose not to send her to Hogwarts, they were obliged to find her some other kind of magical education deemed acceptable by the magical government (the Ministry), whether that was sending her to a boarding school in another country (expensive) or paying for her to have at least ten hours per week of tutoring from a person deemed by the Ministry to be sufficiently qualified (even more expensive, and most registered and qualified tutors preferred only to take students from certain families). It was simply not permitted for a muggle-born child to go untaught: the risk to the Statute of Secrecy was deemed too high, and while it was technically possible to bind a child's magic and make them and their parents forget they had ever been told magic was real, that was a last resort, and the effects on the child's long-term mental and physical health had never been formally tested, but were thought to be severely negative. And yes, wizards and witches were capable of erasing memories, but it was only practised in extreme circumstances by qualified professionals, as necessary, and binding or draining another person's magic was very strictly illegal unless it was the government doing it.
The professor hadn't seen fit to mention these possibilities earlier because she hadn't thought they would ever be relevant: Hogwarts was really the only sensible option, it was natural that Hermione should go, and since she was extraordinarily unlikely to fail her end-of-first-year exams, it hadn't seem worth going into the contingencies for that event, or correcting surrounding assumptions. (The Drs Granger were rightly very unsettled by the way the professor had seemed to feel it was more important to placate and humour them than it was to give them an accurate representation of the true situation they and their daughter were in.)
Little though Hermione's parents cared for the idea of any child of theirs leaving school at sixteen (shudder), they decided that that was the new contingency plan; Hermione could perfectly well take GCSEs as a home-schooled student at an independent test centre. (After the professor left, they told her grimly that if she decided she would rather pursue a career in the normal world, rather than the magical one, she would be sitting a full range of GCSEs, at least ten and hopefully more, with OWL scores not mattering beyond those needed to keep her wand and their memories, and then she would be leaving Hogwarts and attending a nice sixth-form college and then university; whereas if she decided she wanted to move to that other world full-time, she could focus on her OWLs and only take a few GCSEs, perhaps three or four, just in case.)
Hermione herself was extremely eager to enter the magical world, confident that she would belong there as she never had among ordinary children, hopeful that she might get to make friends other than demons; but she appreciated the value of having a back-up plan, and the prospect of sitting what would effectively be two lots of GCSEs was far less scary than the prospect of living, month-to-month, constantly frightened of being taken out of school. Her parents had said, when she had first started yoga classes, that she'd only be allowed to keep it up if it wasn't having a bad effect on her marks at school or her progress with the violin, and the first six months had been very stressful for her, until she passed her Grade 4 music exam with Distinction and it was decided that her schedule must be fine. (And Grade 5, that she was working towards now, was roughly equivalent to a GCSE in Music, so that was one that would be out of the way at least.)
While she did feel a little guilty for upsetting both her future professor and her parents with her questioning, she decided that the end had justified the means. She really had needed to know those extra details about the legal position she was in, and how the society she was about to be introduced to worked. Knowledge was power, after all, and conversely ignorance was both weakness and danger. And Professor McGonagall didn't seem to be resenting Hermione for having prised the information out of her - the wide-eyed innocence and air of enthusiasm were every bit the good tools Forneus credited them as, more so because they came naturally - rather she seemed to feel guilty about the ways the society she lived in was less than perfect, and also ashamed of having been caught misrepresenting it as better than it was, and eager to make amends. That was extremely useful, as Hermione doubted she would otherwise have agreed to take them to Diagon Alley the next week, rather than waiting to do school shopping until the official supply lists came out. (Most of the set texts stayed the same from year to year, but a few subjects did vary; there would be a second letter coming at some point in July or August with the official list, at which point they might need to return to the Alley to purchase one or two extra books.) After all, Professor McGonagall had a full-time teaching job with additional pastoral and administrative responsibilities. She must be very busy. In gratitude, Hermione didn't even press her as to why it was that some subjects had consistent teaching from year to year, and at least one didn't. She'd find out soon enough.