There's no way at the moment to test a website using espresso or uiautomator via the standalone browser i.e. chrome.apk on Android.
You can open a chrome.apk and inject a website URL using uiautomator, but you can't interact with it enough, to make testable.
You should probably choose other cloud device lab provider, for example:
They all support Appium:
I've just took a look into firebase test lab and they state that
Firebase Test Lab is a cloud-based app-testing infrastructure
So picking firebase test lab as a framework for testing web-sites through the well-known browsers seems to be a wrong choice since you need to have the access to the application code that you're going to test.