Your registered nurses are the product of the modern world. They didn't exist 200 years ago, let alone a thousand.
That was a simile, a metaphor, nay, an analogy.
And while "advanced practice registered nurse" did not exist thousands of years ago, midwife did, which is a form of APRN. Physicians, surgeons, bloodletters, herbalists, apothecaries, trepaners and all forms of medical practioners, both useful and quackery existed.
As for our fantasy world, casters who require less training and study than wizard exist by RAW.
- We have the racial caster, who has a cantrip or a 1st level spell from bloodlines, like the aasimar, tiefling and some elves. They should count on the list as part of the spellcasting population but its not something you can choose for yourself.
- There are people who have become Fey or Shadow Touched, getting a 1st and 2nd level spell. ("Survive a date with a nymph" is risky but it just might work to get you some mojo without years of college. Or you might get drowned, who knows.)
- There is a feat specific to a failed wizard; Magic Initiate, giving them 2 cantrips & 1st level spell. There are multiple flavors of Initiate for the tone deaf bard, irreverent cleric, or half-blood sorceror or a not-warlock who managed to keep some power without signing a pact.
- Ritual Caster is another feat that provides spellcasting. It isn't as punchy as a cantrip but being able to cast a 1st level ritual spell like Unseen Servant 5-6 times an hour can prove very impactful, essentially letting one person run a store or a farm without ever breaking a sweat.
- Eldritch Adept Invocations can include casting spells (mage armor, speak with animals, disguise self, etc) with no warlock levels or abilities.
- Spellsniper feat teaches someone an attack cantrip, making them a spellcaster.
- There are the lesser casters in Tasha, the Sidekick Spellcaster. "The sidekick might be a hedge wizard, a priest, a soothsayer, a magical performer, or a person with magic in their veins." They are distinctly less powerful than a full caster and yet are spellcasters.
Ultimately, any 4th level characters of any class, race or attributes could have some spellcasting without being a PC class. If you use Tasha's variant humans rules, any 1st level human farmer can, by RAW, be a spellcaster.
They could be a failed apprentice (Magic Initiate, Ritual Caster), have a "no lie, there I was face to face with..." story (fey/shadow touched), been trained as an assassin or mystic warrior (Spell Sniper), made a sacrifice at a forgotten temple and received a boon (Eldritch Adept) or hundreds of other scenarios.
Most of these are PHB feats so even if you discount TCoE, the partial caster is still a base game RAW feature.
