Sometimes, basing a ruling on the "spirit and intent" of a rule can be a slippery slope. We weren't there when the rule was written and we didn't speak to the people that wrote it. How certain can we be that we truly know the "intent" of the authors?
But on this one, I'd be confident that neither the spirit nor the intent of the rule was broken. The catcher not returning the ball directly to the pitcher is, I believe, intended to prevent unecessary delays in the game. The catcher wasn't delaying the game- the batter was by running when she shouldn't have!
Calling a ball here would be on overly officious technicality. But if you really want to get technical...
There is a list of times when a batter may leave the batter's box between pitches. If she leaves when not entitled to, the umpire may issue a penalty strike. Yes, one of the exceptions when a batter may leave the box between pitches is "on a swing".
But running all the way down to first base is not, in itself, "a swing"! It is a separate action and one not exempted from the "penalty strike" rule. The rule says that the umpire may rule a strike if the batter delays the game for
any other reason than one of the listed exemptions.
Tell the coach that you accessed the penalty strike
before the catcher threw the ball, so his batter was already out.
Is that technical enough for ya?
(Disclaimer: The above is not an official interpretation published by any sanctioning body I'm aware of.)