Dark Briar Barrage is extremely powerful against 3 family types. Don't get splinter bomb, it's completely outclassed.

That's the point of the Shivermist, stops enemies so your teammates can hit them. Besides the instantly refreeze 80% of the time. Although Volatic tempest is a combo of Ash of Agni and Shivermist so you could try that. IMO get Ash if your looking for damage and Shiver for crowd control or VT for both. Keep in mind Shivermist takes eliminates knockback.

One other thing, The vortex bombs are made for crowd control not damage.

The shield doesn't really "counter" anything. The defenses on the shield only applies to defending against sources of fire and shock.

"the mad bomber set reduces some status resistances, and the owlite shield counters the the shock and fire reductions"

You ignore how shields work, I already pointed it out. Good luck dying.

Full Mad Bomber is the very first set of 5* armor I ever owned, and I didn't even start making other armors until I had nearly every bomb in the game. I've done runs all over the place in Full Mad Bomber. Learn to be cautious, or learn to love giving heat to your teammates, but it is the easiest way to become a bomb damage machine. And since you are ignoring vortex because "lack of damage" then I think Mad bomber is exactly what you want.

As has been stated a few times above, you don't understand how the resistance on shields work. They have no impact on how much you burn/shock. What it means is that if you shield while in fire (or zombie fire breath), it will take more hits than if it didn't have the fire resist. This is still handy, but it's not doing what you think it's doing. If the fire/shock is a problem for you, you have two choices: 1) Don't use Mad Bomber 2) Pay for good resist uv's on it. I think Medium will bring the negatives back to zero, but I'm not 100% on that.

Also as stated above, Vortex isn't generally used for damage. In Zombie Hordes Electron Vortex can crank out nice damage, if you learn how to chain it to bunch them all together. But really it's not for that, it's for crowd control. And teammates who use charge attacks (Brandish, DA, Blitz, even alchemer users, etc) will LOVE your vortex. Really pumps up their damage.

When it comes to status bombs, it's really hard for me to stick to just one. AoA is a good all-rounder, but utterly useless (and sometimes dangerous) on a fire level. Shiver, when spammed, can be a great help to an overwhelmed party (noobies, FSC, LOA, etc). Voltaic Tempest is fun. And when solo, chaining AoA and VT together melts slimes to pieces, I can clear levels without ever using a direct damage bomb. And don't forget poison! It makes the final wave of an arena an utter joke, as the menders go ahead and murder all the monsters for you.

Before you choose shards over DBB, test out the shards. They are a totally different bomb now, and you may not care for it. While DBB "looks the same" as nitro, it has some key differences. 1) It doesn't blue wipe your screen, it's gentler on the eyes 2) it doesn't knockback, which can annoy your teammates, but instead tends to knock things down, especially wolves and gremlins 3) It mows down beasts at a delightful rate, and does pretty good on fiends and gremlins as well.

To sum up:
I think what you're suggesting is good. If damage is what you want, I would suggest adding an elite boom trinket so you have max damage with mad bomber. Then, do nitro, dbb (because of what I said above), AoA, and Dark Retribution. If damage plus team friendliness is what you want, I would do DBB (they will hate you for nitro), Electron Vortex, and swap out AoA, Shiver, VV, and VT depending on the level and the team.

If you wanna roll Mad Bomber, then I'd say learn to love the vortexes. Mad Bombers survive by not getting hit, which is achieved through a combination of control and weaving, and (short of shivver), the vortexes offer beautiful control for bombers, as well as more damage than shivver can output. Drop a vortex, suck them in, drop a 'splosion, let them die. Don't assume bombs aren't effective because they don't deal huge damage; especially as a Mad Bomber (whereupon the low damage is increased) you do want to consider more than just killing fast, because, frankly, sometimes "kill everything asap" just isn't an option.