ci: speed up CI by installing fewer playwright deps

[espy]

Overview

run-test.sh does npx playwright install --with-deps "$CLIENT", which is quite slow on GH CI. This PR is an attempt to avoid --with-deps and selectively install only the few things Playwright actually needs, according to this guide.

Testing recommendations

Run the CI on this PR and compare it to the CI runs for #9205, this one should be faster, especially on the browser-adapter tests, some of which (eg. this one) take over 5 minutes each (sometimes much more), most of which is installing Playwright deps.

Specifically, compare the Usage pages of the two runs, those show the cumulative time of all runs at the bottom.

Checklist

• I am not a bot
• This is my own work, I did not use AI, LLM's or similar technology for code or docs generation
• Code is written and works correctly
• Changes are covered by tests
• Documentation changes were made in the docs folder