Why?????

Are you shooting the rig enough to wear out the factory extractors to begin with????
Hence either extractor, the rifle is still not going to be a controlled feed action, and you will wear out a sako extractor in about the same time you wear out the factory extractor, meaning that at about the same round count, your going to replacing the extractor again.

Truth is, just replace the factory extractor for $15, and be done with any extractor problems for decades more of use/ remaining life of the barrel.
Note, the rem extractor is only good for a one time install.  If you pulled the extractor to clean behind it, then you need to replace it with a new extractor isntead.
[youtube]https://youtu.be/5jG-rkGjc_k[/youtube]

And don't get me wrong, since a sako extractor mod has it's place in 700 actions that are going to see high rounds counts from target shooters that will get the bolt face hot enough time and time again to de-spring the factory extractor metal / A bolt face extractor channel that is worn out and factory  extractor will not longer work, but for a rig that may only see 1K rounds a year, the Rem factory extractor is more than needed instead.

Hence lathe cut the old extractor channel out, braze in a bushing to get the ID back to the correct size, mill and drill the needed channels, then fit a new sako type extractor next.
Note, when you do wear out the sako extractor, you will need a new one fitted as well.

Explain, since your just taking about a 223 rim case, and would dare to guess  that the maybe a worn out extractor lip, or one that was removed and it reinstalled if there was barrel work involved/Someone pulled it to do detail cleaning and tried to use the old extractor isntead.

So on that note, would just pick up a New model 7, 223 extractor, install it, and see if it solves all your problems.
Worst case, your out under $20 before you have the bolt mod'd to a sake extractor.
Again, you need the 223 extractor, since the 300blk out uses a 233 case as it parent case.

https://www.brownells.com/schematics/remington-/model-seven-sid816.aspx#r15sid816