The "computing" that's closer to pure math is algorithmic theory, cryptography, and computational engineering, which is an extremely small subset of the field.

The majority of what computer science practitioners do is general purpose coding, which doesn't have nearly as much to do with the underlying math.

Or, in other words, how proficient would an applied math person be at building a front end? And how many of their skills would they be able to leverage?

That's the overlap, or lack thereof.