That depends on whether you can trust the browser. For example, browsers have long had flags to indicate whether they’re being driven by webdriver, but you can simply recompile the browser without those flags.
is it really just "simply recompile"? I would assume some intensive patching and learning would be necessary, especially for someone who isn't familiar with the source or the build process.
Sometimes even more simple -- there are some methods people use that are as simple as "copy and paste this javascript that overwrites some properties". There's a lot of people scraping the web, so there's somebody out there that has done the work for you already.
My point is, you don't really know what software is connecting to your web server.
If you're automating a browser, you're probably technical enough to compile a patched web browser -- or, at least, to use someone's script to compile one, or to download one that someone else has built.