If the gate to the DBT is only in Home World where the Dream Children are, then we have an added complexity -- only after encountering clones of Crono, Lucca, and Marle does a DBT gate open up in either CT:DS or Chrono Cross.
Not necessarily. You first encountered them in the Dead Sea and the Gate didn't "appear" until you encounter them last at the end of the game. I think what is more likely is that it was always there, but just inaccessible until you had the Time Egg from Belthesar.
And as far as the signficance of it being in
both worlds in New Game +, I think that is just to access certain non-canon endings and should not be read into too much.
So you're left with the following evidence: The Gate is only in Home World and probably was there all along since the time of the dimensional split. In CTDS the Gate only appears after going through the 3 dimensional vortexes, and the gate appears in the Bucket to 1999 AD which just so happens to be the time period that the entire future was sent to the DBT.
So I think: A Gate to the DBT appears whenever a severe alteration to the timeline occurs, and in the location that catalyzed that alteration. Now, what if dimensional travel is a pre-requisite for actually
accessing the Gate to the DBT?
That would have very interesting implications for Chrono Cross, as it would imply that only Serge and co. could go to the DBT and defeat the Time Devourer since they met the prerequisite of being dimensional travellers, and would also perhaps explain the need to split the dimension in two in the first place, further elucidating Belthesar's plan.