There isn't a straightforward answer since the Pokemon series tends to have wildly different depictions depending on what source you're referencing (anime vs. core games vs. other media). However, since your motivation is ultimately to write fanfiction, it isn't terribly important what the "real lore" is since you are inventing your own anyway. So this answer isn't going to list every single time any Pokemon has ever had any role in medicine, just some ideas on where to start with inspiration.
Chansey and Blissey have consistently been depicted as being associated with Pokemon Centers; the Pokemon anime even shows a nursing school exclusively for Chansey. Their Pokedex entries describe their tendencies to assist the sick and injured, with the Legends: Arceus entry referring to Chansey as "doctors' doubles". Wigglytuff is a Johto Pokemon who is depicted in the anime as being the staff of Pokemon Centers in Kalos.
Most Grass-type Pokemon in general can conceivably be imagined as having some kind of medicinal properties. In particular, Venusaur and Meganium are known to have aromatherapeutic qualities, though as mentioned in sumwun's comment, Aromatherapy is also a move along with several other moves that restore HP. The Pokemon world canonically includes herbal medicine items too.
Psychic-type Pokemon could be viewed as potentially having healing properties due to their sensitivity towards and ability to control people's mental and emotional states. For example, despite its threatening reputation, Hypno has Pokedex entries describing it as assisting doctors with insomniac patients in hospitals. Espeon is described as desiring to protect its trainer from harm. There are better candidates outside of Kanto and Johto, but overall, this could be an avenue for exploring fringe/experimental treatment modalities or issues relating to mental health. You could also include any Pokemon with a reputation for being cute or friendly (which is subjective!) as being emotional supporters in line with real life therapy animals.
At the end of the day, you can honestly just take inspiration from any real life practice even if it's not present in the Pokemon series. Folk medicine in the real world has a rather messy history of involving all sorts of animals (turtles, tigers, etc.) and often includes unconventional systems involving energies or powers. So if you really want to include a certain Pokemon, I'm sure there's some way to get creative about it.
Miscellaneous potential sources of inspiration:
- How ingredients derived from Pokemon are used to craft TMs in Scarlet and Violet
- Historical life in the Hisui region from Legends: Arceus (for your "starting from nothing" theme)
- HP-restoring items
- Berries
- Check out some of the spin-offs, like the Mystery Dungeon or Ranger series—these tend to have some unique takes on the Pokemon world
As for whether Pokemon moves can be used on people, the episode you're thinking of is EP027 - Hypno's Naptime, where it is stated that moves can have side effects if used on humans. There are definitely more positive outlooks on the subject—for instance, in Legends: Z-A, you can help a man fix his back injury with Grapploct's Octolock. As with everything else, it's ultimately up to you how you want this aspect to work in your story.
