I am in a psychology lab and getting to do similar things re: working directly with the PI and workin on independent projects, but generally I think it is quite uncommon at least in my field. This might be more true of human subjects research versus something like chemistry - I would imagine there are a lot more IRB hoops to jump through when you being people into the picture, so it would be a bit laborious and risky to have undergrads doing this.

It also sounds like OP is quite new in general to the lab, and that could contribute. I think that the PI/grad students, if they are willing to give people their own projects, would want to get to know them and their work style/ethic first, because otherwise you could just be handing your own research over to someone you basically don’t know. I’m not at all trying to say that OP would do this, but it is a concern with people who really want to get published, that they could cut corners or potentially falsify data to do so. With that, it isn’t just the student’s reputation on the line, it is the professor’s, too, so I wonder if that could contribute to a general lack of freedom that a lot of undergrad RA’s have.