Thing is, angles don't really have units (the technical term is they are dimensionless). They are a length (the subtended arc of a circle) divided by a length (the radius of the circle). When you want to do something like get a sine wave of period T, you inevitably have to include a 2π somewhere.
Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular frequencies ugly and unintuitive to work with), I think this notational trick would've been very useful!
Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular frequencies ugly and unintuitive to work with), I think this notational trick would've been very useful!