The goal was more certainly correctly allowed, but unlike the Manchester United goal were De Gea went down, there was definitely an opportunity to pause the game here before Liverpool were in a goalscoring position.
Ultimately, these situations are impossible to legislate for satisfactorily. Fortunately, there wasn't a serious injury here, but it's well within the realms of possibility that it could have been (this being an example where an injury can occur, despite the situation not being in the 'obvious' category). Strangely, the player who hit the ground harder was the defender who got back to his feet - and that landing did not look pleasant. Equally, it would have been interesting to see the wider view if both defenders had stayed down.
I don't envy Dean's position, and ultimately there was nothing he could do once play continued.
His positioning for the 3rd Liverpool goal, however, would be highly inadvisable for an official at any level. It's a great clip for discussion on decision making and positioning. That nothing happened here is down to sheer luck rather than by design. Dean could have easily been skelped with the ball, or there could have been a handball decision on the Newcastle defender. In the latter event he would not have been able to make a decision.
His position was almost borderline blocking the Newcastle defender(s) ability to play the ball.
Obviously VAR would have been available, but that would not be a recourse available at lower levels. An excellent clip to help with education of less experienced officials.
Clip is at 2:29, the later angles really show how awkward this was.