There is an assumption here that the 4I is founded by the Regency. It is not.
Indirectly, the Regency is involved in the formation of the 4I, and indeed some of its brightest and best do journey there, leaving behind the failing Regency.
Democracy cannot work over parsec distances with J6 coutrierrs. There is no FTL commo or J6 breakthrough. Therefore, democracy is not an option but for a small state.
The Regecny is a brave experiment, but ultimately doomed to failure. It needs to do too much in a short time, while its problems and internal divisions multiply.
Of the successor states to the Regency, some follow its better ideals, some go their own way and one rejects everything the Imperium ever stood for, yet manages to drag itself down into darkness anyway.
The Fourth Imperium is founded on the concept that the only way to govern over long distances is through trustworthy local leaders owing fealty, allegiance and honour to their far-off superiors, but who are trusted to do what they must within a framework set down by the central commnaders.
This was the ideal of the Third Imperium, and it worked. Once iot became corrupted, it failed. The Emperor Search is simply a quest to find someone that is worthy to lead and rule; for a long time the 4I answers to a council of advisors and a Vacant Throne.
Two pivotal events change that - the emergence of an unlikely paragon in the last hours of the Dominate War, who shows that sometimes central leadership is the only way, and that even the worst people aren't totally evil - and the discovery of a person deemed fit to ascend the Vacant Throne.
The 4th Imperium is smaller than the third, and in many ways it is the core of what the Imperioum should have been - burned clean, purified in the dark days of the Reconstruction, and born of the actions not of generals and admirals, but of diplomats.
History records the founders of the Fourth Imperium as visionaries, but they were not warriors. They were peacemakers.
Their ideal was the man who stopped fighting to sdave his people - Strephon. Though he had more reason than anyone to fight on, he stopped because that was all he could do for his people.
It is that side of the Imperial Nobility that the 4th Imperium reveres. People like Duke Craig, who also refuised to fight for rulership, instead choosing to look after his people. Those are traditions to be proud of.