Monte - the freebore diameter on my various .223 Rem reamer prints is 0.2242" +/- whatever the manufacturer's tolerances are. All are based on my original 223 Rem ISSF reamer from PTG, with various freebore lengths. Every single one cuts a chamber so tight that when I use a Hornady OAL gauge, the bullet stops when it first encounters the [start] of the freebore. It requires a much harder push on the plastic rod to get the bullet moving again, albeit grudgingly, until it encounters a second [harder] stop, i.e. the rifling. It makes determining CBTO and COAL at "touching" much more difficult than with my .308s, but it is still doable with a little practice and close attention. I still can recall the very first time I took a measurement this way on my first .223 Rem F-TR bolt rifle set up for shooting 90 VLDs. When the bullet hit the start of the freebore, I thought it was actually touching the lands. So I tightened up the set screw and pushed the assembly out with the cleaning rod. Needless to say, I was a little surprised to see the bullet seated way deeper in the case neck than I thought it should be. Once I figured out what was happening and got the measurement working as described below, the difference between the two measurements was very close to 0.169". Imagine that.
The primary concern is that you have to push on the little plastic stick much harder when the bullet first encounters the freebore, but then still be able to feel the rifling and stop a short distance later when the bullet actually encounters the rifling. Like I said, it takes practice and care, but it can be done. There is at least one way to "check" as an internal control whether you have actually accidentally pushed the bullet well into the rifling. I use a cleaning rod inserted from the muzzle to push the bullet and OAL gauge back out as a single unit to take the measurements. If the bullet has actually gone into the rifling any appreciable distance, you can feel additional resistance when using the cleaning rod to push it back out that is not present if the bullet is merely just touching the lands. Further, I make a "Measurement Set" of 8 or 10 bullets grabbed at random from every Lot # of bullets. I number each bullet on the base with a Sharpie, then measure and record the OAL. The Measurement Set goes into a labeled bullet box and is used until that Lot # of bullets is finished. I use the average of the measured CBTO and COAL values, which are also an internal control in that you can immediately see when one measurement is much different than all the others and re-measure with that bullet, if necessary. An additional internal control is that the SD of the COAL measurements taken with the tool should be very close to the SD of bullet OAL. If it is not, then you probably introduced additional error into the measurements when taking them. I attached a copy of a recent set measurements taken from a new barrel on my original .223 Rem F-TR rifle, if you want to get a feel for what the results look like.
Again, this approach with the Hornady OAL gauge is far from ideal due to the tight .223 Rem freebore, but it can still be made to work successfully in the event you are unable to get Alex's method working satisfactorily with your .223s.