To force the account picker, append the following alongside your client ID and redirect URI parameters when you send the user to GitHub to sign in: &prompt=select_account.
While the user is still shown the prompt if they have multiple accounts, it allows users that don't have multiple accounts signed in to quickly add a different GitHub account and sign into Modrinth with that account instead.
A simple change to the auth system that lets you pick your GitHub account following this change: https://github.blog/changelog/2024-06-07-account-picker-updates-for-oauth-and-github-app-sign-in/
While the user is still shown the prompt if they have multiple accounts, it allows users that don't have multiple accounts signed in to quickly add a different GitHub account and sign into Modrinth with that account instead.