"The central idea behind Remote Browser is that there's no need to reinvent the wheel when modern browsers already ship with an extremely powerful cross-browser compatible API that's suitable for automation tasks."
People still call it 'selenium' for whatever reason, but chromedriver/geckodriver/edgedriver are all implementations of the protocol for each existing browser (safari's might be built in?).
Granted the web extensions API would likely allow for some additional powerful options, but at the cost of some missing features.
Selenium is just a tool built on top of the WebDriver API. One of its main disadvantages is needing to run a complicated proxy program (like geckodriver, ChromeDriver, etc.) built individually for each browser in order to drive your instance. As a result, users sometimes suffer from hard to debug edge cases and other pain points.
They also make interacting with JavaScript on the page a bit painful. For example, injecting JavaScript into the browser with Selenium can be quite an ordeal [1], so you're somewhat limited in what you can do by what Selenium's developers decided to focus on. It also complicates deployments by adding another moving part to the overall equation.
In contrast, the Web Extension API is now part of all major browsers, and makes interacting with different page contexts effortless. To give a sense of the project, we wrote an interactive tour of Remote Browser which runs browser instances on our backend.
You're writing that users suffer from edge cases from the individual webdriver implementations. My experience as someone working with gecko and chromedriver on a daily basis is that the number of edge cases stemming from browser behavior (such as moving to, clicking, and focusing on elements) is a much more frequent pain than differences in the webdriver implementation.
Thats already what the webdriver API accomplishes, and its a w3c standard. https://www.w3.org/TR/webdriver/
People still call it 'selenium' for whatever reason, but chromedriver/geckodriver/edgedriver are all implementations of the protocol for each existing browser (safari's might be built in?).
Granted the web extensions API would likely allow for some additional powerful options, but at the cost of some missing features.