I am sure others will chime in but no, you don't generally add a muzzle brake to a bolt action rifle. A proper crown is more important for accuracy. You are more likely to use a bipod and since you have to take your hand off the rifle to cycle the action anyway, recoil has less of an impact on your follow up shot.
I mean, it might make it look cooler though and if you really want to piss off the guys next to you at the range you could put a miculek style brake on there. Either way, it isn't likely to help your accuracy.
I am guessing that your 5.56 was a semi auto though right?
If you are going to shoot a lot in a given period of time a brake is great. If you shoot precision rifle competitions, you use either a brake or a suppressor.
I took a long range rifle class at Sig. My .308 loads were fairly hot (175gr SMK at 2720 fps). All shooting was done prone with the gun along the centerline of the body. By lunch everyone's collar bone was sore.
I shot 120 rounds per day over 2 days. And anything that reduced recoil would have been appreciated.
A brake on a bolt gun is not about fast follow up shots. Its about
1) the cumulative effects of recoil on your accuracy and followthrough.
2) having the rifle stay put enough so you can continue to track your target in your scope after the shot breaks. This isn't just for shooting at things that walk. Its nice to see the steel swing if you hit it. If you have a good scope and its bright out, you will be able to see almost as well as someone with a good spotting scope.
I got a chance to take a shot with an Accuracy International .308 Win Mag. It had a brake that would kick up dust for 20 ft off to each side. But it recoiled less than my 12 lb .308. A lot less.
Don