In Choices of One, also by Zahn, Thrawn states his belief that "while other voices can advise, there must be one voice to decide, to rule, to be obeyed." In other words, Thrawn believes in the order and discipline of a monarchy rather than a democracy, which is possibly a product of his interactions with Jedi and Chiss politicians before joining the Empire. As such, his main goal, defending the galaxy - and the Chiss in particular - from external threats, primarily the Yuuzhan Vong, is something he thinks would be best served by the Empire under his rule, at least militarily. He seldom displays interest in power for its own sake, merely in the order that comes from a clear hierarchy of power; this is hardly uncommon in soldiers, who tend to despise anarchy in all its forms.
No doubt, if Thrawn had been alive in 19ABY - and he may have been, as Survivor's Quest hints - he would have left the New Republic alone, correctly deciding that his efforts were better suited strengthening the Empire of the Hand, the Imperial Remnant, and the Chiss Ascendancy, leaving the New Republic alone rather than wasting his forces against it. The Empire of the Hand built by Thrawn's successors, Admiral Parck and General Baron Fel, was described by Fel's son, Chak, as "more of a confederacy than an Empire in the strict sense," and there is no evidence of the use of force to prevent internal dissent in the new Empire.
Thrawn's philosophy andthe New Republic's philosophy were not as different as is often supposed; he is the benevolent despot, the New Republic the liberal democracy, and Palpatine the tyrant. The Prussia of Frederick the Great as compared to the Germany of Merkel, as compared to the Germany of Hitler. Which makes the Old Republic the volatile and useless Weimar Republic in this analogy. Lucas would like that.
As long as the New Republic didn't descend into the anarchy and infighting of the Old, there was no particular reason for Thrawn to go after it in 19ABY; in fact, it would be better to prop it up as potential buffer to the Vong. But in 9ABY, when The Thrawn Trilogy took place, internal political infighting was already beginning to tear the New Republic apart, with Borsk Fey'lya playing political games that threatened the polity's very survival. It also controlled only a tiny portion of the galaxy compared to the glory days of the Empire and less territory than that controlled by Thrawn after merely one year of campaigning. If the Republic couldn't survive a year against Thrawn's weakened Empire, it certainly couldn't survive a full-scale Vong invasion.