Unless you are going all the way to put together a match quality rifle, the factory action will be just as accurate. Now a custom action will function much nicer. Smoother, tighter which gives the perception of performance.
However, I have played with many commercial and surplus actions and have had no issue getting these rifles to shoot as low as the 1's. That's average group, not best group.
My basic standard is 5 rds in 1" at 250yds (1/3MOA). Then capable of hitting clays at 700yds (roughly 2/3MOA). Then a MOA size gong at 1000yds if the cartridge/bullet is big enough to warrant shooting that far.
That will not win you many BR matches but certainly bring home the venison.
The key was the barrel and how well the chamber was cut. Then there is how well the stock is designed, bedded and trigger set up. From there load development and optics. Garbage in, garbage out.
So if you want to drive them into one hole that you need calipers to measure, and have some extra cash, go custom. At the BR level, it can make a difference.
For LR hunting and varminting, the weakest link is the bullet we feed these rigs to function on game. There are no hunting bullets that can shoot below the 1's. Most will not shoot below 1/3MOA. I have read one really big name bullet was only QC'ed to 1.5MOA (that has sinced changed but...). You will see no benefit in using a custom action here. Except the Bling factor.
In fact, none of my actions were trued and all that other BR stuff. They still shot very, very well..for a LR hunting rifle.
Jerry