But the withdrawals didn't force them to admit they had taken a massive loss, the withdrawals forced them to take the massive loss at all. It's not a loss until you sell, right? They sold to cover withdrawals.

Apparently they had $48B in withdrawals in a one-day period. Trying to imagine any bank that wouldn't need to take losses (to the point of being potentially insolvent) in order to deal with that. Yes, obviously SVB was still very poorly hedged given current interest rates, but they probably could've unwound their position in a much, much more favorable way without the run, to the point where it's possible they could've done so without ever being "insolvent".