NOt sure if you're interested in just random brainstorming, but here are a few ideas that have mainly investigative or RP uses, rather than practical healing applications.
Most of this is just off the top of my head (caveat emptor, I'm very new to 5e!). However, a few are from 3pp bestiaries: "remove tumor" was related to some sort of attack form; and "treat the slowed condition" came from a weapon attack implicitly causing a mobility-hampering leg injury. The haircut thing is a nod to the old "barber surgeon."
DC 5
DC 10
- extract a tooth
- give shave / haircut to order
- treat minor / cosmetic (0 hp) injuries (eg, rashes, burns, scratches)
DC 15
- medical communication (eg, read journals, give lectures, draw anatomical diagrams)
- craft simple prosthetic (eg, dentures, non-functional hand, fake eye)
- preserve biological sample
- treat a wound to avoid scarring
DC 20
- craft complex prosthetic (eg, spectacles, basic jointed hand, realistic fake eye)
- remove tumor
- extract rot grub (within 3 rounds of penetration)
- minimally injurious torture (!!)
- deduce general medical history upon examination (eg, congenital issues, major childhood diseases)
Varies
- extract rot grub (more than 3 rounds after penetration)
- treat / calm madness
- treat injury-related condition (eg, blindness; slowed by hamstring, sprain, fracture)
I'd like to see Medicine checks applied with abilities besides Wisdom, if that rule is available. Seems like forensics and academic uses are good match for Intelligence. A competent, stylish haircut is totally a Charisma check, because no one wants to offend the king at his daughter's wedding ball.
Also wish there was a Physician's Kit (with, eg, blades, pliers, painkillers, tweezers, sutures, jars, etc) that would somehow synergize with Medicine). But it's not obvious how to do that, given how Ability Checks and Proficiency work.
I like all of these. They fit the ability and the level of non-magic use. Especially prosthetics because it's not an angle I was thinking of but very appropriate.