How do you do cross browser testing in JavaScript library development?

I am developing a JavaScript library. I want to check if the created library works properly in multiple browsers by the integration test.

In this case, how is it appropriate to carry out the integration test?

I'm assuming the following method.

1. Establish a project for testing in a repository different from the repository where library development is done.

2. Therefore, create a sample application including the library under development (the library is privately distributed with GitHub Packages etc.) and deploy it to the test server with basic authentication.

3. Write the test code using a tool such as TestCafe. (The test code is managed in the repository of the sample application.)

4. Run tests in multiple browsers.

How do people who normally develop JavaScript libraries perform integration tests? I would be grateful if you could teach me how to manage integration test code, tools used, and tips.