If it were about the FSD implementation, things would be very different. I'm pretty sure that the FSD is designed to handle cross traffic, though.

I do not trust normal users to understand what beta means

Yeah, Google kinda destroyed that word in the public conciousness when they had their search with the beta flag for more than a decade while growing to be one of the biggest companies on Earth with it.

When I first heard about it, I was very surprised that the US even allows vehicles with beta self-driving software on public roads. That's like testing a new fire fighter truck by randomly setting buildings on fire in a city and then trying to stop that with the truck.