Refactor LoginButton to use useMutation for async sign-in

[fhennig]

Context

In website/src/components/auth/LoginButton.tsx, the authClient.signIn.social() call is handled with a raw .catch():

authClient.signIn
    .social({
        provider: 'github',
        callbackURL: callbackUrlThatDoesNotImmediatelyLogoutAgain,
    })
    .catch((error: unknown) => {
        showErrorToast({
            error: error instanceof Error ? error : new Error(String(error)),
            logMessage: `Login failed: ${getErrorLogMessage(error)}`,
            errorToastMessages: ['Login failed. Please try again.'],
        });
    });

Problem

This is an async call that doesn't follow the established pattern of using useMutation for async operations. Using a raw .catch() means we miss out on proper loading/error state handling that useMutation provides, and it's inconsistent with how other async calls are handled in the codebase.

Suggestion

Wrap the sign-in call in a useMutation hook for consistent error handling and to align with the existing pattern.