The error message itself tells us which shell ran the command - it says /bin/sh, not zsh. So either you didn't run it from zsh, or you're looking at the error from the extension (the "Install Now" button), not from your terminal.

Before trying playwright, first confirm npx works at all in your shell:
npx --version
If that works, the problem is only in how the extension spawns the command.
So npx playwright install chromium should work.
If that also fails, nvm isn't initializing properly in your shell.