Cross-browser testing

## Description

Integrate a test feature to perform cross-browser testing for the Leonardo web UI.

Possible resource: https://github.com/microsoft/playwright/blob/master/README.md

## Why do you need this feature?

Currently there is nothing in place for automated browser testing. We are limited to manual testing in supported system browsers and within VM. The process is inefficient, and would be nicer to have some degree of testing to show any bugs.

## Additional context

Start with the linked Playwright README and inspect the Leonardo web UI's existing development and test setup. Define which supported browsers and user flows are in scope, then add automated cross-browser coverage whose runs demonstrate that the selected UI flows work across those browsers.

Written by the indexing model from the issue text.