Should I weave in why I chose the college I’m applying to into my essay or is that not necessary?
Your Common App essay need to be generic, since it is used for all colleges you apply to. And not really about why you want to apply to any college, that is not the point. If a college has a supplemental “Why College X?” essay, then yes, you will want to talk about the college by name.
If the college does not use the Common App, then it depends on their question, I guess.