How would that be "totally incorrect"? There are only 24 hours in a day. It's incorrect to claim that there's 25, 26 or 27 hours in the day - even if it's useful and proper, it's not technically correct.
In the context of the train departure time, it's easy to work out what is meant by a weekday 00:37 arrival, but if they were divorced from each other for some reason, weekdays at 24:37 would be easier to interpret.
Being awake past midnight brings up a few challenges with calendars. My watch wil count a late (late) night walk against tomorrow, not against today. As a human, I consider "tomorrow" as "after I've gone to sleep", not "after midnight". 00:37 is still very much tonight.