I asked this question long ago, but wanted to see if I can get some new perspectives this time.
Why is the radians implicitly cancelled? Somehow, the feet just trumps the numerator unit. For all other cases, you need to introduce the dimensional analysis / unit conversion fraction, and cancel explicitly. Is it because radians and angles have no relevance to linear speed ($v$), so they are simply discarded?
In the 2nd equation, you are simply multiplying radius by how much angle it sweeps per minute. 2 times $360\pi$ somethings. No need to describe what there are $360\pi$'s of? $360\pi$ seems meaningless without the units to describe what you're counting (radians)