I'm not a barrel maker or an authority on it.I've just done some reading..and had a career in the machine trades.
At least Hart,Shilen,and Lilja are button rifled.
Badger was cut rifled,Obermeyer is,Kreiger...maybe Bartlein? not sure.
They have all made some famous barrels.
Per what is written by barrel makers discussing this topic,as has been mentioned,a button forms the rifling...as a thread forming tap forms threads,in a chipless process.
The barrelmakers say button rifling must be done before contouring,as there is a certain amount of "ostrich swallowing an orange" effect.Variance in wall thickness will result in variance of the rifling.
The size and surface finish of the bore before the button passes is critical.If you have done press fits on dowel pins,you know .0005 more interference makes a big difference in force required,plus if there is nowhere for the steel to go,if the bore is undersize,galling is more likely.I have seen the smeared cheese look of galling on the lands of a modest price barrel.
Some modest priced barrels do not have all te reamer marks lapped out before rifling.The bore has tool marks as if it had been cut with a boring bar.If the bore is a bit loose,the button does not smear the tops off the tool marks.Have you ever seen lands that look like a file?I have.
OK,all this is about cheap barrels.A precision,lapped hole makes it work better.
The "30 second vs 90 second" claim may apply to production barrels,but I suspect at Hart and Lija,a bit more goes into it.
So,yes,now there is stress in the barrel.Stress relief involves heating pretty hot,approaching critical temp,then cooling.Remember the bore,the rifling,is finished except for lapping.
Now its time to profile from the rough blank.Maybe stress relieve again.
After all this,its time to finish lap.
On a cut rifled barrel,the outside can be profile turned,and the bore drilled and finish reamed before stress relief.The rifling cutter puts no stress in the barrel.
IIRC,Kreiger double stress relieves and cryos before rifling.
Cryo and the heat of stress relief can effect the steel surface,so,post stressrelief,the bore is lapped,then rifled,then finish lapped.Nothing else happens to the bore.IMO,the cut rifled barrel creates a "fresher,sharper" bore.
I don't think hammer forging can produce quite the precision and definition ...the sharp,clean corners of the other processes.And,the barrel is inherently stressed by the process.You have to decide between work hardening and stress relief.Stress relief will ,I think,negate work hardening
I think hammer forging excels for producing military and general hunting barrels.The chamber can be formed at the same time,a nice throat,it can all be chromed or nitrided.I think a Daniel Defense bbl is crackerjack for a three gun or battle rifle.
They are all good for intended purpose.
I want a Kreiger for an AR I have...and,could be I'll only be semi-satisfied with less.BUT,aWhite Oak is significantly less money...
The bug hole tack driver is a joy,but,in reality,I'm not all that good of a shot!!
If I get an honest 5 shot .750 MOA rifle most of the time with some 1.00 MOA bench groups...That won't cut it in Barts Competition world where the standards are higher..but it will outshoot ME!
I'd probably trust a White Oak to do that.
I'd be disappointed with a Kreiger,maybe,if I couldn't get some pretty small groups.5 MOA? Maybe.But,from target center to bullet center,the difference between a 1 MOA rifle and a .5 MOA rifle is .250 at 100 yds.Its the radius .
Typical sandbag bench shooting,I doubt I hold that close.
FWIW,my fake 1903A4 and my Garand both have Criterion barrels on them.
They both shoot real well.It does not prove anything,but last time I put it on paper,checking sights before antelope season,I put a 12 in bull on a target at 350 yds.With my 2 1/2 x Lyman Alaskan,I used a6 oclock hold with the post reticle.Slung up,prone,left hand resting on my rucksack,3 shots went into just under 3 in,center of the bull.
As I said,proves nothing.But,no glass bedding,military stock and trigger,combined with my mediocre shooting,I won't complain.