As much as I want to believe that Biden would want to "eliminate" his opponent - Trump, I don't believe he had anything to do with what happened last Saturday.
Sure, it would be so easy to just "assume" that since both Biden and Trump don't like each other, I don't think it was the cause for it. There is absolutely no proof stating that Biden wanted to have Trump killed -or at least, shot at.
However, the Democrats are desperate! They are afraid - as they should be - that Trump stands in their way. They are afraid that if he (Trump) becomes President again, the Republicans will very much take the House and Senate once again. He (Biden) won't have that, and he - as much of a cognitive of a decline he is - will vow to stay in the race, and fight it out. Even though, he really has no choice - but to step aside (when that time arrives). He knows his days are numbered. He may as well, just hand Trump the "keys to the White House" and be done with.