This isn't how physics works. When people talk about dimensions in physics they don't mean dopplegang universes.
They're talking about the concept of the universe being something like folds within folds. What looks like a grid of X Y Z coordinates is believed to be something like origami, where rigid structures are built from folds in space. If you google tesseract cubes on youtube, you'll get a great idea of how we have a xyz grid folding in on itself.
For example gravity already "warps" space. That isn't a theory that is how it works.
A great way of imaging it is of a 2 dimensional piece of paper with a rock at placed on the paper. the force of gravity drags the piece of paper to the edge of the table, the paper drops off the table onto the ground which is the 3rd dimension. The mass itself get's stuck on hand rail at the edge of the table and is continually dragged by the paper that is continually moving off the edge of the table. After a while enough rocks get caught against the hand rail that it bends and sling shots some of that mass out of the well as light rays. This of course requires the particle mass to be bent into light which exists in what we can consider another dimension. This is why light is considered to have dual characteristics. As both a wave and a particle.
When enough rocks start piling up in the same location, space bends at the sub atomic level, nuclear forces etc become bent and result in a fission/fusion reaction. Obviously I'm paraphrasing something that is super complicated, but it help illustrates the concept. What we see as XYZ isn't what is studied by actual physicists.
It isn't a theory that time and space is getting warped into higher dimensions, that is a requirement of the universe/gravity existing, or at least a common theory believes this to be true.
It isn't at all unrealistic to think we'll find some trick to exploit the interaction between
https://en.wikipedia.org/wiki/Fundamental_interaction
These forces represent higher dimensions and we know they are directly involved in regularly folding what we imagine is 3 dimensional space.
There's no reason we couldn't figure out how to manipulate these forces in the near future to allow for ftl.
I'd also add the fact we are even having this conversation is why trek needs to be updated.
Around 2010 we as a society went through a mini enlightenment, youtube allowed for people to get educated in mass as free entertainment. In the last decade we've experienced a boomerang of a mini dark age where people have become more ignorant. There's no reason we couldn't use trek to reverse that trend.
You're modestly moving the goal posts, I despise FTL in general. I'm not saying it is a given, I'm not saying it is likely. I'm saying star trek relies on the premise of what if we can figure it out.
I'm also not suggesting that Trek needs to be diamond hard sci fi.
The idea we have to go between black and white diamond hard sci fi or just have writers doing whatever they want with the trek universe is farsical.
There's a middle ground, don't include more magic than you need to create a solid franchise/story.
A lot of what people have been saying in this thread, is that it's happened in the past, therefore it should continue to happen.
They give zero reason why it is integral to the story. They also don't seem to mention the difference between episodes that require it and episodes that don't.
This isn't the 1960s trek can move on while still maintaining it's best ideas.