There are advantages to tight neck chambers, but nothing is free.
There are advantages to tight clearances in racing engines, and most modern factory engines. But cause this without a plan and you're likely to see the opposite of advantage.
".001 thicker on one side of the neck how much runout would that bullet have?"
Your BULLETS would have far less runout(often too low to measure without a polish).
Loaded ammo TIR as measured off exposed bullet bearing (in this case) can still be well below .001, even with .001" thickness variance. Those pre-seating with mandrels see this in benefit of that action.
An advantage of a low clearance condition is less working of brass necks. This is especially useful where your annealing is less than perfect, as with tighter & tighter clearances the need of annealing for consistent tension diminishes. It can even diminish to zero need for annealing(with zero need for sizing).
Another advantage to low neck clearance is quicker neck sealing. This, reducing carbon back flow that often builds to a carbon ring interference. Combine low radial clearance with low chamber end clearance, and you'll never need to concern yourself over carbon build-up in the chamber neck.
So far these attributes help with lower ES.
Lower neck thickness reduces tension, which is reducing variance of it.
Combine turned necks with tight clearances and you will normally produce (with firing) near nothing for runout from necks. This is good, but if you then muddle up the necks(and/or the rest of the case) with a bad sizing plan, increasing TIR, the necks & seated bullets can be put in high chambered tension. This is where tight clearances are not good for everyone, and where switching to higher clearances may improves results. just don't dismiss that it was your muddling up of this that led to it. And those producing and managing straight ammo will not have this issue with tight necks.
Back to the whole straightening part. In no way does neck turning 'straighten' necks. Only fire forming can do this. And the only way those turned necks will stay straight is if you don't mess that up with your sizing plan.
Probably seems that I'm pro tight clearances

But only where I have the plan and control needed to be successful with it.
Where I've chosen this path, I've been very happy with results.
I also have no problem with normal clearances. With most guns that's what I have. There are just no tight neck advantages with these.