I'll agree with a slight change: The league should bury this unless Ohtani was betting on baseball. In any form or under whatever complex machinations. They draw a bright line at that, post it super prominently in every clubhouse, and have held that bright line for over 100 years, up to and including excommunicating a popular and charismatic would-be HOFer over it. There is nobody who is unclear on that point, or what the penalties are for it.
As long as he wasn't betting on baseball, MLB should be helping him with his defensive strategy to keep the Feds off him, and I agree the league shouldn't care. Betting with illegal bookies is malum-prohibitum, not malum-in-se, in my eyes. It's not great, he should maybe still get a slap on the wrist (or else their deep-pocketed gambling partners will get angry at them essentially helping "The competition", I suppose), but it doesn't call the league's integrity into question. If he was betting on baseball, though, they'd need to react really strongly, or else it's a green light to everyone else that they don't really mean what they say on the signs, and the league really really can't open that Pandora's box. Not even Shohei Ohtani is valuable enough to them to be worth the downsides of crossing that line.