Exactly. If you're having to wring necks then you are not putting your pattern on the birds well enough. Could be because your choke is too open (meaning you are trying to shoot them beyond the range at which your choke is effective), or it could be you are simply fringing the bird with your pattern.
Pheasants are usually pretty tough targets, unless you get a stupid one that tries to fly into the wind. it is not usually easy for most people to put the pattern on the bird well, but if you do 6's will kill them even at very long range (as when our partner wounds one and you need to finish it at 60 yards).
However, at 40 yards all (or almost all) the #5 pellets will blow completely through a pheasant. There might be a few underneath the skin on the far side (because of the elasticity of skin) but there will be none or almost none in the meat, which is nice.
The problem is that most people are not able to shoot well enough to use a tight choke, and without a tight choke you've got a pretty sparse pattern at distance with #5s.
I've seen guys try to have their cake and eat it too by buying 3-1/2" magnums loaded with 5's or 4's, but then they usually start flinching and missing because of that.
My advice is, if you had to ask, then use 6's with something like an IC choke and figure out how to hunt so that you don't have to take long shots on unwounded birds. Often that can be as simple as everybody being very quiet and, if there is wind, moving into it, not with it.
Very good shooters will do fine with tight chokes and #5's, and it gives them the ability to reach out and take birds most people should not even be shooting at.
4's are overkill. You never need pellets larger than 5's to reliably kill pheasants. You are just driving your pellet count down for nothing.