I would say if they can grip the ball or a bat it's not too cold to be outside.
I think I will ask my shoulder surgeon his opinion. He should know how cold is too cold for a HS pitcher to throw . I'm not saying that 37 is too cold, it might be, it might be fine with proper dress and warming up enough. But early in the season, why take the risk.
So I should not wonder or question the coaches decision, just leave it up to his expert opinion.
Perhaps you thought I was being flippant - I wasn't. I just don't think that 37 degrees is that cold. I grew up in the northeast and played HS games in weather colder than that. I think the fine motor skills (fingers, hands) will fail before there's an injury risk to the major muscle groups (assuming proper warm-up and clothing). Call me old-school, and by all means ask your doctor for an opinion. And don't ignore the risk of not throwing enough before the start of the season.
My opinion, and it could very well be wrong, is throw in the barn , get quality work in, prepare for the season, and if it's 40 and sunny, then warm up well and throw outside... if its 39 or colder, cloudy, wet, and windy it's not worth the risk...
I have read some replies from coaches that state they need to get used to the colder weather for the the early games, but honestly where I'm at by late Feb the weather is not an issue, so we are talking about 3 weeks... not really long enought to merit throwing in the cold to get adjusted to the cold , so why throw in the cold ?
I see your point about the fingers and hands getting cold before the arms and shoulders, but why do I see schools in the north coming south to play? It's not for money... it's because they do not want to play in the cold.
If you have the option to throw in a barn I would certainly take that over going outside. As far as schools in the north going south, we did that because fields were playable and it was a fun trip. And yes, it was warm.