The Old Republic only stood for as long as it did because the Jedi were taking care of enough problems to keep it from falling apart, by taking them out of the picture Palpatine ensured that any government that tried to replace him would eventually fail because they wouldn't have an army of mystic space-ronin to keep them out of a collapse spiral.
Basically, yes. The Republic post-Ruusan Reformation is so loose and weak so as to not offend local and corporate elites, and they only lasted so long because an army of Übermenschen psychic space knights served as their ambassadors, diplomats, peacekeepers, and generals. The Jedi mollycoddling the Republic stifled its maturity and growth; they didn't bother to learn from past mistakes or work to create a more effective government because the Jedi are just there to patch the dysfunctional one so that things keep on going.
The KOTOR/SWTOR era Republic is a mixed bag, since while there was corruption within it, to the point where someone like Revan would choose to wage war against it instead of just running for chancellor, it was still strong enough to make a difference. The Republic Army and Navy kept things safe for the most part, and the Jedi merely acted as the tip of the spear, akin to the Space Marines in 40K's Imperium of Man. They're powerful, but even without them, the Republic is capable of defending itself to a certain extent.
The Republic under the Pius Dea cult was a military powerhouse even without the Jedi, conquering millions of worlds and de-fanging alien powers, and during the Mandalorian Wars, when the Mandos were a powerful, galaxy-threatening civilization instead of a gaggle of has-beens with overpriced armor, the Republic forces were able to hold off the Mandalorian Neo-Crusaders for a while despite suffering heavy losses, until Revan was able to amass his army of Jedi Knights and defeat the Mandalorians. And of course, we can't forget the special forces of the Republic Army which was at least strong enough to give the Sith during the SWTOR era the occasional black eye, being able to keep up somewhat with the Jedi despite their lack of psychic powers.
As for Palpatine ensuring that any government will fail without him, that's only for Disney-canon. In Legends, the Empire did have a succession system in place, but Palpatine's advisors were so weak that they failed and the Empire regressed into warlordism, and with Darth Vader dead, no one was strong enough to stabilize the mess. If Inquisitor Jerec succeeded at the Valley of the Jedi, he might've been strong enough, but Kyle Katarn killed him. It's telling that if Kyle does follow in Jerec's footsteps and takes in the power of the Valley as a Dark Jedi, he DOES become Emperor, and he's in a position to keep the Empire running smoothly.
Same thing goes for Thrawn; if he didn't rely on the Nohgri for protection and succeeded at defeating the NR with enough conventional battles, he could've claimed the title of Emperor and restored Palpatine's Empire. At least, until C'baoth turns Thrawn's soldiers against him with the Force. That, and Palpatine did have clone bodies on the side, so he was ready for the possibility of death. And even with his final death, that didn't end the Empire for good.
Galak Fyyar tried to seize the throne by assembling an army of Dark Jedi with Desann and going on a crusade to wipe out the Emperor's hated Jedi enemies; if Desann succeeded, his army of Dark Jedi could've been used to pressure the rest of the Empire to accept Galak as their new leader. Same thing with the Cult of Ragnos crisis; if Marka Ragnos was fully revived instead of just half-assing it by possessing some Dark Jedi, then he'd be strong enough to force the rest of the Empire to bow before him, and probably strong enough to crush the NR fleet in battle, given how powerful the Ancient Sith were at Battle Meditation and making suns go supernova. Even Naga Sadow had to wait for Ragnos to die of natural causes before he tried to stake a claim on the man's throne.
The Empire that Palpatine left behind after he died was at least stable enough due to its top-down command system, that it eventually grew back into full power in the Legacy Era, and even some Jedi saw the advantage with a more stable Imperial government, since some Jedi chose the Empire over the Galactic Alliance and even ruled the Empire for a time. When the Empire's former Jedi enemies wind up joining it because it's more stable than the alternative, that's saying something.
The only thing that I would've added would be the restoration of the Imperial Senate, so at least there'd be a voice for the commoners, acting as a counterweight to the Moffs, not to mention that the Senate can elect a new Emperor in the odd case that the current one croaks, instead of just having the Moffs wait for the strongest thug around to stake a claim on the throne. The last thing any civilization needs is to copy the flawed formula of Byzantium, where coups and palace intrigues led to one disaster after another, from the loss of Asia Minor to the Fourth Crusade.
