Keith is heartbroken and upset Shiro’s gone, and he can’t just move on. When the team says that they need a new leader, Keith keeps reminding them that Shiro is gone. It hurts him more that Shiro is gone rather than Voltron not being able to form.
So when he says “You wanted me to lead Voltron? This is how I do it”, he’s bitter and pissed that the team has made him become the leader even though he’s mentioned time and time again that he doesn’t want to lead the group to Shiro because Shiro is the leader and he never thought that he would actually pilot Black because he never thought he’d lose Shiro again.