Your on the right track. 223 definitely needs to be reamed. I have both the .309 and .311 reamers and use have settled on the .311 , because I wanted a little bit thinner neck.
I only shoot cast bullets, so it allows me to use a larger reamer because of going a tad over the slugged barrel diameter. I have a number of CZ52s and TOKs and a .308 reamer would be too small for all of mine, as with a larger diameter bullet, the neck would still be too thick to chamber good.
If you're using the LEE die set, you can usually get a good enough crimp on the bullet with the factory crimp die for that first firing. After that, you might be OK.
Another thing you can try is to run it back through the sizing die after reaming. sometimes I have seen where some brass may actually open back up in the neck in this process due to the brass being thicker there when you size it after trimming, and because brass is usually harder in the main body of the brass where you are getting the new neck from. The thinned brass sometimes will size down just a bit.
I have found the Forester trimmer is the ticket for me. I made a small device that I use to measure and mark the brass for cutting, then I use a small tube cutter and a Lee Zip Trim with a 3 jaw chuck to cut it with a small tube cutter and also use steel wool to polish it while in the zip trim. Then I size it and then into the Forster for trimming and reaming all in one shot. Then back into the zip trim for a quick champfering with a small Lee champfer / de-burring tool, and walla, a nice piece of brass in less than a minute.
You're definitely on the right track. Just keep messing with it until you get the end result you need. Keep us posted if you will. I've never used .556 brass. Is there much of a difference? If I get some, your info might help save some messing around if there is a bit of a difference.
