But the referee may not have been actually looking at the offside line at that moment. Like any human, he can only look at one place at any given moment, and he has no opportunity to go back and look at what he just missed. That is what the TMO is for.
The TMO is in a booth with multiple screens.....
... which allow him to see multiple things at the same time. He can even have his assistant synchronise the replay streams from two camera angles, for example, a ball carrier grounds the ball in-goal but his trailing foot touches the touchline. The in the front on view of the grounding, the player's trailing foot cannot be seen; from the back on view, the foot can be seen touching the touchline, but grounding cannot be seen. Syncing the two video streams allows the TMO to move back and forward frame-by-frame to see which happened first.
So how does this all help in this case? One camera on the offside line, the other on the ball and TJP picking it up.... synced streams allow the TMO to determine that Lawes was offside before TJP lifted the ball.
At the speed of the modern game is played at, the referee doesn't a hope in hell of making that judgement accurately.