I imagine you could still run it off an HDD, it'd just take forever to load. In game everything is in RAM though, so unless you have less than the recommended RAM or have a bunch of apps in the background, it'll be be just as playable (if you don't care about long load times)
Problem being, many games are starting to use more streaming-type continuous loading for seamless open world games, and those are definitely going to be affected by the lack of an SSD.