Go ahead and turn the necks all the same if you like. They may crack on the first firing. Bottom line is you are going to shoot poor brass in a factory chamber to see if you can get your handloads to shoot more accurately. The only gain you might get is a more consistent neck release.
If some necks are .014" and others .010" the cases are most likely thicker elsewhere. This will mean the internal dimensions will be different. That difference might negate any tiny gains you might see with consistent bullet tension.
I would suggest you read the information from 6mmBR.com
223 Rem + 223 AI Cartridge Guide
I suggest you scrap your idea of tweaking the rem brass and buy some Lapua brass. To quote the article I linked:
"To achieve high levels of accuracy in the .223 Rem, you need quality brass. Among the dozens of .223 Rem/5.56x45 brass makers, Lapua produces the highest quality and most uniform brass we have found. If you look at the charts, new lots of Lapua .223 brass showed the smallest extreme spread in weight among all brands..."