That's not an exaggeration, no. It used to be 10% but it's just gone off the scale recently.
Marchmont is studenty (but less so than it used to be) but you're unlikely to find a semi there - it's largely tenement flats with a few exceptions near the old sick kids.
Grange is very not studenty but expensive - your budget might get you an upper villa flat.
Stockbridge is quite chichi but also busy and expensive. Orchard and craigleith bungalows are walking distance but not right there, if you don't want a suburban feel.
Trinity has the right housing stock but not the naice local shop/amenity thing going on.
Edinburgh council catchment area maps are your friend for schools - double check as they can change midstreet.
Morningside, bruntsfield, Blackford and Grange primary school catchments have just changed to accommodate the new school on Canaan lane.
If you end up in catchment for boroughmuir, be aware that the school isn't actually in its own catchment (it's a bus ride away for a lot of kids in e.g. South Morningside).
Greenbank might suit you - 10 minute walk down to Morningside, good schools and in budget.
I think you're going to struggle for what you want (4 bed period semi) where your want it (in town with vibe) as that just isn't the prevalent housing stock where you're looking. Consider terraces and flats/upper/lower villas and you might have more joy.
Also, seems daft but if you're going to be spending years in and out of hospitals your North/South decision might want to factor in whether you're traveling to the Western or the ERI.
Good luck.