It's actually the opposite.
Being the first to Printing Press can turn the tides for both religion and world congress, so is being the first to Acoustics.
Babylon is also the only civ who can pop the pre-medieval wonders on deity since they get there fast enough, with other civs the moment you research the req tech other civs already finished the wonder it enables.
I'm not sure what difficulty you play on but in Deity the early game decides the rest of the game.
You get ahead? You win.
You fall behind? Others will fund congress and possibly embargo you, pop fundamental cultural wonders before you, and ultimately run away like the Road Runner.
I play on Deity. This is not true. It's not the opposite. The start didn't get more important in BNW, it got less important.
Now, being the first to Printing Press WILL turn the tides, but that doesn't mean it helps you any more than having more late-game science and not hitting Printing Press until Pisa is gone. You can also reliably build one of more than half of the per-Medieval wonders without being Babylon.... it's the Medieval wonders that you can't get. So, not sure about your comment there.
You can run away with both civs, quite easily, with any start, as they are both very overpowered. Touting Babylon's start is pointless, since you should be evaluating the overall science output of the two civs over the course of the game, balanced by their security. The turn times for finishing games for these two civs are very comparable, so let's not overstate the importance of the early game for science.
I got a turn 254 Deity science victory once with Korea in BNW, where I delayed the NC, only had one wonder: PT, was in the tundra so bad growth, didn't get a religion, and had ZERO rivers/mountains in any of my 4 cities (one city had a lake). Think of the turn time it could have been if that game was at all a decent start....
I would argue that on a random start, Korea actually has a better start than Babylon, if only because it has a coast start bias, so it gets guaranteed 2x benefits from food routes. Babylon has no start bias, so it's pushed behind the 10 civs that do have it for coast. This is very bad on Pangea, and still not so good on balanced land/sea maps.
As for # of specialist slots decreasing in BNW:
Food routes lets you grow your cities, on any start, while working specialist slots and planted GSs. Korea on 4 cities may have lost 6 specialist slots overall in BNW, but now you can put 3 more in your NC city = 9 specialists worth of science, which means you've effectively only lost 3 overall specialist slots, and zero if your NC city happens to have a mountain. Overall, Korea's going to be able to better use it's UA than in G&K, even though the total # of slots has decreased.