The New Monarchy still wants to dissolve the Consensus, to my knowledge. In fact, at one point they sought to place Zavala as the "Leader of Unimpeachable Character" that would rule the City as its Sovereign, under their banner. Zavala declined, of course. The Vanguard seems stoutly loyal to the Speaker, so this isn't surprising to me. Ironically, I think his declining the position speaks to his character, as the New Monarchy sees it.

So the old superhero adage: Only trust power to those whom don't want it...

Basically. But the problem isn't such that he "Doesn't want it" but still has it thrust upon him. Zavala doesn't want it, and won't accept it, so he can't be made to accept it. I believe this theory we've speculated with above, may have actually been the point in which the New Monarchy suggested Zavala as their Sovereign. Surely they wouldn't have cast a vote to dissolve the Consensus with the Concordat, knowing there was a chance they may actually, finally succeed, without having a replacement leader in mind, right? So when it came time to cast their vote, I suspect they probably suggested Zavala as the heir-apparent, at the same time. In one of my more "Tinfoil-y" theories, I speculated that the New Monarchy's leadership may have actually let Lysander live, and escape the City. The reason for this, in my theorizing, is that the New Monarchy may believe that Lysander's methods might be needed again, or maybe even that the New Monarchy would see Lysander, himself, replace the Speaker.