The idea of changes to the past propagating forward necessarily involves a second time dimension -- there's no avoiding it. To be perceived, "change" necessarily involves a time dimension against which the change occurs. When what is changing is events in time, then you need a second time dimension for it to change in. You can't observably change the past in a 4D universe!
If we confine ourselves to our usual four dimensions and try to imagine a change to the past (and its consequent effect of its future), without a way to step outside of those four dimensions, we can't speak of change to the past at all. We remember how Lincoln's second term in office was spent touring the country, North and South, to heal the wounds of war and effect a lasting reconciliation. We remember building a time machine to observe his great series of speeches and how careful we were to avoid doing anything which might contaminate the past. And ?suddenly? we remember his assassination and the botched Reconstruction that followed it. How can that be? If Lincoln died of an assassin's bullet 150 years ago, then our memories from earliest childhood are of that death and its consequences. And if he lived to an honored old age, then our memories from earliest childhood are of the Grand Old Man himself.
A few writers have appreciated that -- a particularly good example of one who did was William Tenn in "The Brooklyn Project". In that story, the Brooklyn Project was to prove that the present does not change even though the past is changed.
After an introduction, the chief scientists is ranting about people who think that changes to the past change the present:
“As you know, one of the fears entertained about travel to the past was that the most innocent-seeming acts would cause cataclysmic changes in the present. You are probably familiar with the fantasy in its most currently popular form—if Hitler had been killed in 1930, he would not have forced scientists in Germany and later occupied countries to emigrate, this nation might not have had the atomic bomb, thus no third atomic war, and Venezuela would still be part of the South American continent."
He goes on a bit more and then presses the button to send back a a series of probes. After each probe, things change more -- more radically each time -- and finally, when the experiment is over:
“See,” cried the thing that had been the acting secretary to the executive assistant on press relations. “See, no matter how subtly! Those who billow were wrong: we haven’t changed.” He extended fifteen purple blobs triumphantly. “Nothing has changed!”
Bottom line: Asimov had to do something like what he did if he wanted to have the people in the story perceive changes to history.
BTW, I agree with your criticism of how he did it: He wasn't being terribly consistent but was focusing on the story. (Which IMO made him a greater writer, not a lesser one.) So to answer your question, In a 4D universe, I don't see any way you could have The End of Eternity.
For an example of someone who did a better (IMO) job of handling the perception of changes to history, look at Poul Anderson's Time Patrol series. In that series there is only one 4-D space-time, and when one changes the past everything futureward of the change changes. The only way to retain your memories of the "former" timeline is to be to the past of the change. Nicely done. (He's not the only person to use that idea and I doubt he's the first. But I do believe he's one of the best.)