While the P-223 3x might not be my choice, it might serve you well for your intended use. One thing to do: Go to a box store where you can play with one.
Check to see if it will focus at the closest distance you are likely to shoot the rifle, and whether the field of view (amount of area around the target you can see) is acceptable. If it does, it would serve you as an intro to magnified optics and you could then evaluate whether you want anything different later.
I took many deer in my early years (I'm 65) with a rifle with a fixed K4 Weaver 4x scope. I did almost miss one at stick throwing distance that ran right under my tree stand, because all I could see at 4x was deer hair and I couldn't tell which end of the deer the crosshairs were on! Fortunately, he stopped and let me move my scope around enough to find a reference point at the base of his neck.
For this reason, I've been a fan of low powered variables for general purpose, such as hunting versions of 2-7x33 type with simple relatively thick duplex type reticles with a finer center cross hair. I keep such scopes down on the bottom magnification except when extra magnification is needed. I do hunt with 3-9x40mm scopes almost as much, and down at 3x it is not unlike the P-223 you are considering.
FWIW: If you are not going to exceed 250-300 yards, you do not need a BDC reticle or exposed turrets. Your AR with the 50 yard battle zero for your scope will shoot with a straight hold on out to almost 300 yards, with the bullet being no more than about 3-4" from point of aim at any point. A regular budget priced hunting scope in the 2-7x range might serve you well. Leupold, Weaver, Burris, Vortex, Bushnell and Nikon make such scopes.
In fact, the "P" in the P-223 stands for Prostaff. The capped turret Nikon Prostaff 2-7x32 might be a better choice for you at not much difference in cost. Would certainly do better up close and reach out better for a longer varmint shot.
Just check field of view and close up focus capability of any scope before buying.