Missing 10% crit on neck, not to sure about barb builds but i would spreadsheet it to see whether keeping the 93% crit damage or rerolling 10% crit chance would be better. Your weapon has str/vit and if you're trying to boost your dps you might want to throw it in a weapon reroll calculator to see how much better it would be rerolling the vit to 10% damage or rerolling the base damage. doesn't look like you've rerolled the weapon yet.
Like I said the only "glaring" thing i can see looking at your profile is the lack of 10% crit chance on neck, having crit chance on your neck slot is pretty mandatory if your after high dps since it's the highest you can get out of one slot. That's why I suggested finding a spreadsheet to see whether the crit chance would be better than the crit damage. At high amounts of crit chance, crit damage will start to out weigh crit chance but only if you're at or around 50% crit chance.
If you haven't gifted your weapon yet, you should reroll the socket into 10% damage and then gift the socket back onto it.
Even though it's already been made clear, the reason why CC on amulet is so important is because it's over-allocated compared to elemental damage. Directly compare it to bracers, which can roll 20% elemental and 6% CC, whereas amulet can roll 20% elemental (the same), but CC goes up to 10%
I see nothing wrong with wearing that set, mind you. It is fairly powerful, but, as with any other, you need to work on it. Not having at least lvl 60 gems invested into upgrading it via caldesann's despair in cube is plain laziness and has no excuse. You need about 1h to level one L gem 1-60 with empowered grifts with your 800+ paragon main char, or even faster if you have some friends that can help you run grift 60 under 5 minutes. I have no complains on your gear, despite the fact that you do need few ancient pieces, all seem to be more or less as it should. PPl already told you to reroll that att speed on weapon into 10% dmg. You might need more physical resistance to push higher solo grifts and if I may add, I find that war cry impunity is not good as veteran's warning. Good luck.
I notice that armor has been retained on pants and chest, which seems wasteful to me, particularly when armor on chest can be re-rolled to 11% elite dmg reduction, or earthquake dmg. I also notice move speed has been retained on the boots, where ancient spear dmg would likely be preferred (shifting the move speed to paragon).
The gear looks really nice for a pretty big damage boost I would say as many others have that the lack of the double crit on the neck is a big loss but also you can try and aim for ancient damage range double crit rings as this will really boost your damage output. Apart from that and the 10% damage on the weapon you are looking good my friend!
Your hellfire has strength, try crafting one with crit chance instead (ph20%, cc, cd, socket) also your weapon - 10% damage instead of attack speed. What is your GR score? you should look into caldesann's despair in the cube, it boosts damage quite high if you put it in every ancient piece.