Simple explanation:
Any decision by a Rabbi is not only decided by someone accepted by that group, but only gains acceptance or authority if it is accepted by other trusted individuals as well from other groups (ie: cities, tribes, countries etc...). And so the idea is that by using sages from previous generations, we know they were accepted by everyone and have authenticity/authority.
This is the benefit of "decentralization" and no one individual telling you what is right and wrong.
In regard to some other questions you raise:
Jews who follow Torah know it is extremely important and cannot be altered.
However, since Jews were never ALL in one location at ALL times, ie: for much of the history they have been split (either in exile to four corners, or even in Israel under multiple tribes each having its own copy from Moses), this would make the Torah "decentralized".
Then if you compare the Torahs from each group you can see no change (certainly in meaning of text). Thus verifying its unchanged state.
So the Torah and teachings of sages have authenticity to them. As far as them not being altered from the time of writing/decentralization.
Most Rabbis point out that we have a tradition of Rabbi to student (who become famous Rabbis) going back all the way as an unbroken chain. Furthermore often there are more then one chain, for example two famous chains that go back extremely far: Pumbedita and Sura, all Rabbis recorded and have a length of approx 800 years unbroken from around year 200AD-1000AD. But similar chains can be made going all the way back with books devoted to this topic.
Thought experiment:
Imagine post Moses, the tribes did not have one ruler that could make such changes and have everyone agree. In fact the Torah was written into mountains making such a process even harder. Afterward when kings were introduced we quickly had two kingdoms. Even then, where Kings or people followed idolatry it was only short bursts between idolatry and Judaism - no chance for a major change.
In general Torahs were not altered, only burned or names of Hashem removed - this is documented.
We know these Kings are insulted and treated as rebellious Jews, and yet I know of no record where they attempt to change the Torah. Even so... would every person change it, would the other kingdom relent to such a change?
Even after the first Temple when the Jews were exiled, by this time the Torah would have spread and been less under control of one leader. You would have two Jewish localities, Babylonia and Israel - where two sets of Jews not under one ruler would remain.
Eventually the septuagint is written in around 275BCE making future changes nigh impossible since now non-Jews had copies. Here is the one time intentional changes to the Torah for a non-Jewish king were made, it was miraculous because they all made the same changes (even though they were separated by the King to ensure an accurate translation). I believe only 10 minor changes are made - totally superfluous to your question. And those changes were not accepted, and are recorded.
One can postulate "what ifs" and they are hard to absolutely refute because there are few facts. I think it is more intriguing if you could provide a more factual based question like "at the time of ___ what stops the alteration of a Torah" etc... the more facts to the argument the more interesting it becomes. Also to properly treat the general question a book of information is needed, and yes they have been written, but I have gone on for long enough I think.
Specifically when analyzing King Saul, King David and King Solomon (1070 BCE (monarchy established) and 930 BCE (Kingdom of Judah established) there is also the distributive aspect.
In fact I actually think that Judaism has a form of government similar but different to America's President/Court/Congress. Judaism has King/Prophet/Court. Neither has complete control of the over though it is clear they can try.
So for Saul he had a clear counterbalance of Shmuel who in fact caused the throne to move to David, and Saul was powerless. He did not and could not rewrite the Torah to his advantage, and even he had a whole city of Kohanim killed, but that would certainly be beyond his reach. And perhaps not in his interest as he followed the Torah in general.
In regard to David he was rebuked for taking BatSheva by Nathan. David did not respond with an ammendment of the Torah. In fact we have recorded that many times David was told he was wrong for example mephibosheth is said to have disagreed and proven David wrong in Torah Berachot 4a. The Tanach speaks of David's sins, had he been able to (ie: desired) he apparently could not even change what the prophets wrote let alone the Torah. So either he could not, or he tried to keep the authenticity. Also famously Gad was also a prophet in the times of David.
In regard to Solomon he had Nathan the prophet and Gad as well in his times. Similarly in Melachim it mentions he had too many wives that affected his relationship with Hashem. We are not told that he undid or changed the Torah, in fact religious observance under him increased.
And also there are the Priests who also have a vested interest in the Torah not changing (they get Maaser, like a tax) and yet the tax payed to them would likely be a first concern for many to change if they could.
There are a number of entities who are a bulwark against each other from changing any laws or story as that would adversely affect them. And these constituents are either large or powerful.
Also there were hidden Torahs like the one kept with the Kohanim and two for the King, one carried at all times (at least sefer Devarim) and one in his vault.
The Assyrians had no affect on the Kingdom of Judah as far as I could find, but the above would apply to them as well since propehts existed then too.
