I think a lot of things can potentially go wrong, though to me some of the ideas in the blog seem less likely than most speculation.

The idea that vaccine RNA can be integrated by endogenous retroviruses is very unlikely, I think. If I remember right, only one family has activity in humans. HERV-K appears to be the only active one. Usually in cancer, or as the link suggested, stem cells (though to what degree in stem cells I'm not sure). Mainly, RNA induced expression systems are used in induced-pluripotent stem cells very often. The expression invariably remains transient, even when researchers don't like it to. Researchers likely would have noticed this, even by accident, at this point, and the experiments have been done to trillions of cells for different experiments over years.

Actually, one reason why this may not happen relates to one of the things I consider a cause to worry about side effects. When your cells is infected with RNA, part of the immune response is the destruction of that cell. It's not all just an antibody-based immunity. Presumably, a cell permanently expressing the spike antigen will be eventually destroyed. The bad thing about that idea is the random cell destruction that occurs could be harmful in a variety of ways.

The other idea about splicing into another virus that was infecting the same cell is beyond my expertise, but I haven't heard of viruses co-opting other elements that are not a stable part of the genome. It could be a freak accident occurrence, but IMO far less likely than the other things that go wrong.