Position for plate umpire?

The PU has to read the play and decide is the ball going to first or coming home. If he thinks it's going to first, he is going to go up the first base line just like there was no R3 but he is going to glance back to see if he is advancing. If R3 is coming home, the PU doesn't want to move too far up the line because there may be a throwback.
Now in your OP he moved up the third base line, why I'm not sure, but if he has a look when the BU comes to him then he can certainly make a call. Now, he shouldn't call the runner safe or out, that's the BU's job, but instead give the BU what information he has and let the BU cange or not change his call.
Also, the manager gets NO SAY in umpire mechanics, NONE! That discussion is going to met with a quick, go to the dugout coach. If he persists it will lead to an early day for him. They can ask about what we saw or how we are enforcing a rule. Umpire mechanics are nothing he knows about, nor something he can discuss.

What I was taught at proschool a little over ten years ago:

PU would clear the catcher, keeping chest to ball and would establish a position in foul territory off the third base line that gave him a view of the pulled foot, swipe tag. With R3 we were taught not to drift up the first base line unless there was a fair/foul call to be made. Even then, the drift would be no more than three or four steps.

Too many things can wrong quickly and you end up with a runner scoring behind your back or getting in the way of a throw to home.

Additionally, Jim now teaches that even though this is not a hit to the outfield, PU still has any play on R1 at third, should one develop from a series of bad throws, or sh!tty baserunning.