If they had simply started the story with him already in the struggling pet shop, and simply introduced his background as a isekai'd pro-wrestler who's stupidly affectionate over animals through an infodump, I really don't think too many people would think "Hey, wait a minute, I need to know exactly what his first six months in this new world were like!"
James Bond films are famous for starting with an exciting pre-credits sequence which has little or nothing to do with the movie's main plot. That pre-credits sequence is simply there to tell you "This is the sort of character Bond is, and this is the sort of action we're going to be showing you him in," and that's what the suplexing-the-princess scene does. It also tells you something about the world, that this is a world where funny takes precedence over strict logic. Logically, suplexing a princess is going to have extremely serious consequences. In this world, those extremely serious consequences are just somehow resolved off-screen and it doesn't even matter how.