npx doesn't run project-installed packages, it uses global downloads
The "npm run" is a command for running defined scripts in your package.json
The equivalent of "npm run" in yarn is just "yarn", for example: "yarn start"/"yarn dev".
What "yarn run" does is allow you to run binaries installed from "dependencies"/"devDependencies".
Say that I run the following:
yarn install --dev nodemon ts-node typescript
I can then run
yarn run tsc --init / yarn run nodemon / yarn run ts-node
And it will invoke the local "./node_modules" package binaries.
This is useful when you want to ensure version-locked, per-project binaries.