Some months ago, I was debating with another poster about how the system requires magic to solve certain problems. I brought up petrification- you get turned to stone, chances are the only thing that can help you is Greater Restoration. They were like, "or the DM could allow the blood of the creature to cure the petrification, or some other means that doesn't involve spells".
Which is true, though by default, the game says "spells only". You have to add in these things, and I'm not sure why, since it wasn't always like this.
In our world, there's a lot of myths and beliefs about the mystical properties of herbs and gems (Gary has a list of these in the 1e DMG, as it happens). People believed that dunking an amethyst into a drink would remove poison from it.
In a fantasy world, there's no reason that can't actually work. The body parts of magical beasts can contain magic that can be harnessed without spells (there was a time when the Monster Manual would call some of these things out, like how "possession of a unicorn horn is a sovereign remedy to poison). Ed Greenwood published quite a bit of information about the properties of herbs in the Forgotten Realms, including some that only exist there, like Trueroot, a universal poison cure. We've had many pages over the years to discussing the mystical properties of various metals and minerals; heck, the Underdark is suffused with radiation that over time grants the races that live there magical powers!
Magical energy can gather in places or creatures. The mere presence of a dragon in an area over time can cause the land to become tainted by it's essence, warping and changing the flora and fauna.
An alchemist could use nothing but knowledge to combine these materials to create "magical" effects without any actual spells involved. But the only thing the game really allows for is the creation of healing potions, for whatever reason.