Is it possible to setup the URLs for profiles as website.com/username rather than website.com/profile/username for example

Yeah, to be very clear about this (what @sudsy is saying), such pages would not be indexable, and would be crappy in other related ways (like, what’s the unfurling image going to be when shared?).

It’s much better to use the same lookup technique on a “real” page. A lot of discussion about this in the following old thread: