No, Saturday Night Live should not move to Thursdays, because it would have to change its show a great deal in order to work on a Thursday show. The show would have to be earlier, and it would have to be more family friendly. People watching television earlier would be expecting a larger budget and they would expect more special effects. The show would likely not work on Thursday.
There's no reason that Saturday Night Live should move to Thursdays. The show is perfectly fine and performing well in its current time slot. Therefore, the network should keep SNL where it is and not mess with a good thing. Other shows can occupy the Thursday timeslot with better results after all.
Its strange to me that people would think that Saturday Night Live, a show that has been in pretty much the same time slot for literally decades, would want it to be changed. Many people have plans late Saturday night, sure, but ultimately, the tradition is worth the time alone.
Saturday Night Live should continue to air on the day it has for decades, which is its namesake. This gives the actors and celebrities time to rehearse their skits for broadcast on the weekend. Additionally, more people are able to stay up from 11:30 PM to 1 AM on weekends, so if it were moved to a Thursday, less people might be able to watch it. Moving it to Thursday would also interfere with the airing of The Tonight Show on NBC.
I do not think that Saturday Night Live should move to Thursdays. I have been watching this show for a long time, and it is so fun to stay up late watching this show. Although some people may be out on a Saturday night, they can always record the show.