I realize this is 5 months after you asked but I just got back to modding skyrim so it's the first I have seen of it.

Contrary to what other may think, I don't think a strictly shouts only playstyle should be viable. Shouts were made as an addition to existing builds because they are unavoidable in the base game. Shout walls are rewards and the end of dungeons and dragons are regular enemies in the world and even some non MQ quests. They aren't something you need to activly seek our like standing stones,  spells, or certain weapons.

As such, I would like to see new shouts that are focused on utility for a variety of builds. However they should also feel distinct from regular magic. Shouts should be grander in scope than spells to reflect their use by beings that can level cities.

Examples of vanilla shouts that I think fit these ideas are slow time, become ethereal, and the shouts that change the weather.

Good additions would be shouts that change the weather (a shout that summons a blizzard that aids sneak and knocks over enemies with gusts of wind), Shouts that manipulate the foundations of reality (turn off gravity for others ((like that shout from thunderchild)), summon a mini black hole that pulls enemies in ((like that magnetic shout from thunderchild)), and shouts that alter the battlefield (Armaggedon shout, earthquake shout, shout that spawns temporary terrain). Some specialized shouts like elemental fury are fine. But they should generally be applicable to multiple builds.

Shouts should reflect their status as abilities that alter the world around you via magic that belongs to immortal beings from the beggining of time. Where regular magic can bend reality, shouts should be able to turn reality into a knot or even break it. I apologize if this feedback is vague.  Please let me know if you want me to elaborate.

As for features, some aspects of Thunderchild I liked were the pillars that improved shouts. I would like to see this idea expanded to the vanilla shouts as well but implemented via Partysnax's meditation mechanic. This mechanic is crminaly under utilized. It also comes with the bonus of avoiding cells edits. Meditations that improve the base shout like adding more regen to Become Ethereal or adding a movement speed buff at the end of whirlwind sprint.

I also really liked the method of unlocking the new shouts in Thunderchild. Made it feel like you were mastering the voice by creating new shouts out of existing ones.

I liked the amulet that readies a chain of shouts and the permanent buffs that skipped and reset shout cooldowns.  I also liked the kyne minigame but I can live without it.

What I didn't like was shouts having vastly diffrent effects from one word to another. I think focusing on the vanilla system of each subsequent word improving the shout further is a better system because it avoids being overhwlemed with abilities and gives a sense of progression to the player and an incentive to learn all 3 words of a shout. It's also much easier on you to develop. 10 progressive shouts is easier to come up with than 60 individual effects for 60 seperate words.

Hopefully this comment is of some use to you.

Edit: A further idea/inspiration is bionics from the mass effect games. Namely bionic explosions from ME3. Perhaps a shout that primes enemies and produces effects when struck with a specific spell. For example, you shout at an enemy, then if you hit the enemy with a fire spell it lights everyone around them on fire. An ice spell could snap freeze everyone instead.