I don't use Selenium, I use WWW::Mechanize::Chrome and Mojo::UserAgent for web scraping stuff, but I appreciate you may already have a body of tests using Selenium.

"Maybe there's a way to expand the list of devices in Selenium::UserAgent"

See devices.json.

"After that, I've connected to my server through the VNC viewer, opened the same Firefox I've been using with Selenium, and there was no anti-bot protection this way. So, that's why I'm sure that I need to use a correct UserAgent and/or some other settings."

Some sites use advanced techniques to determine if you're running headless so your mileage may vary, and sometimes it's not consistent between requests. It can be quite the game of Whac-A-Mole.

"I've set up the Accept header with the code below, but nothing's changed."

At cursory glance I can't help any further help I'm afraid, a deeper dive into the code should prove fruitful.

Update: fix link.