Well, if he's already outed, why not go with the flow? Mysterio's recording can probably be quickly sussed out as being doctored, since he was already well on his way to dying when Peter gave the kill order to EDITH. We see on of his co-conspirators downloading data onto a thumb drive when Peter gets the upperhand of the battle which lends to the idea that Mysterio and his goons used their illusion tech to create the video as one last middle finger to the late Tony and his protege. They get that done and the only thing left for Peter to do is own up to his side-career as everyone's friendly neighbourhood Spider-Man.
Of course, this still leaves his family and friends in constant danger, but really they already were anyway. MCU's New York literally gets hit with an attack once a year, they'll be fine. Plus, without the need to hide, Peter's life could get a lot less stressful. Tony definitely wasn't stressed! Okay, strike that, Tony had anxiety and PTSD up the wazoo, but we have faith that Peter will be able to get through it with the support of his close ones, just like Tony.