Yes those cases have a primer crimp, which has to be removed. Sounds as if you have now managed to do that with the deburring tool. A 45 degree countersink bit, as shown in the video, might work better, and save wear on your deburring tool. However you must be careful not to over-do it, particularly if you shoot hot loads. You may find the primer expands into the chamfer and looks very flat, or even risks perforating, due to the lack of support.
This can even happen when seating the primer, if you use a little too much force.
You won't have weakened the case head, it's the support for the primer (the weakest link) that is important.
There are good reasons why these crimps are used on some factory ammo, they help prevent the primer from backing out or worse, in looser chambers, large headspaces etc, maybe even causing a stoppage, and strengthen it.
I use the Lyman hand reamer tool to do a proper job. Its not expensive
Lyman’s Primer Pocket Reamer removes military crimps from primer pocket. Order today!
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I suggest you also scrape out the residue from at least the bottom of the pocket every reload to ensure the primer, and the anvil inside it, seats flush against the bottom of the pocket, and you get a good solid strike from the firing pin.
When the primer is properly seated the anvil is pushed slightly inside it, which supposedly sensitises it slightly, and makes them more consistent, at least that's what I've read.
The basic Lee tool works fine for this, or you can get special wire brushes to chuck in a drill. Vibratory tumbling alone doesn't get them clean enough IME.
PS: the case on the right looks damaged, as if a hard extraction has pulled up the rim, I'd scrap it.
If using once-fired brass, particularly in .223/5.56, be aware that the internal case volume varies. 5.56 has thicker brass/smaller volume than .223, and is already loaded to higher pressures, so if you used a .hot 223 load in it you could be in trouble, particularly if your rifle is not chambered and proofed for 5.56. Easy way to check is to weigh the cases. Or just don't use 5.56, tempting though it might be.