The problem comes down to the structure of power. Even if we granted that Alicent could accomplish all of this, that she expanded the Small Council, created a Council of all Dragon riders, and created a Council of the Lords, and used these as checks on one another... what on earth does she do to check the Council of Dragon Riders?
You seem to be structuring this with the Dragon Rider Council akin to an executive branch, able to be a check on legislation from the Lord Council who are the legislative branch, and aren't able to make legislation themselves... but what stops the Dragon Rider council from saying "Dracarys"?
You assert the following:
And the only way for Rhaenerya and Daemon to claw back power would be do start a dragon fight and fire a lot of lords, which I feel they would be reluctant to do in peacetime, especially considering that Greens still have Vhagar.
Aegon the First and Maegor already did it successfully. Aegon forged the Iron throne and his control over Westeros without any of the other kingdoms threatening his power and without having any legitimate claim over Westeros other than "I have dragons". Maegor turned a situation up to 11 in bringing about war, and broke the Faith Militant and ensured the Targaryens continued their own traditions and power structure and kept Westeros together. Jaehaerys, as a consequence, was successful as an administrator simply because he mainly only had to administrate, and Viserys coasted off that success.
Furthermore, even if the dragon riders didn't immediately roast the Council of Lords, nothing stops them from threatening them and bullying them into getting the legislation they want anyway. They literally have dragons, they ARE the military
By contrast, look at the USA. Yes, the president is the leader of the military, so in theory they could order the military to do as they bid. The military is made up of the citizenry of the USA, of all those states, so ordering the military to attack their own homelands would be nonsense and soundly rejected. So for the president to get effective legislation through, they must work through diplomacy or political bargaining or reputation. Hard power can't be used within the structure, only soft.
In Westeros however in your hypothetical example, the Dragon Riders are themselves the military, they can do with their dragons as they wish for the most part. It's not a matter of if they will roast the Council members, but when. The dragon riders don't have to use diplomacy, their main power is quite literally their dragons, they can make clear what they want behind closed doors and then have the process rubber stamp their demands as legislation. Hard power is literally built into the system here, so the Council would have to put forth legislation anyway as the dragon riders demand if push came to shove.
If anything, because this power structure doesn't recognize the realities of power within Westeros at all, it's either inevitably going to collapse and revert, or its going to be a mere formality that just does what was originally done anyway.