had intitially the same issue with just npm install yarn@3
This worked for me today on macOS Ventura from the instructions above:
yarn set version stable
# gave
➤ YN0000: Retrieving https://repo.yarnpkg.com/4.0.2/packages/yarnpkg-cli/bin/yarn.js
➤ YN0000: Saving the new release in .yarn/releases/yarn-4.0.2.cjs
# then
yarn set version 3
npm install yarn
tada!
yarn -v
# output
3.7.0
I am now successfully on version 3.7.0 !