It took barely any time for them to think he'd actually pushed Buzz out the window to kill him, that he'd pulled off his arm and that he'd broke into the van to steal car to kill another toy.
Ok so Bo Peep probably believed Woody wasn't a murderer but she didn't have a strong enough voice to override the Pig and the Potato.
Woody isn't innocent either. He clearly has a superiority complex and doesn't think twice about using Car for his own needs without consent .
Woody is the leader but the first chink of opportunity and Pig and Potato are THERE, trying to take over.
It's only through Buzz that Woody learns humility which enables them to bond properly as a group.
Does make me question Bo's taste in men. No wonder she was off being a free toy all those years. Probably part of her therapy recommendations to avoid unhealthy relationships