If you want low self noise out of a mic, Large Diaphragm Condensers are generally quieter than Small Diaphragm condensers, per the laws of physics. I'm not saying all LD's are quieter than all SD's without exception, but within a similar price range, it is frequently true. The flip side is that the mic tends to be bigger and heavier. I ran an LSD2 > PS2 / AD20 > H120 a few times, and it sounded pretty nice. Yes, that's a phantom powered mic, and it's big and heavy. The AD-20 is generally regarded as a pretty quiet pre, and very easy on batteries.
You mentioned the AT4025... I've never heard one, but if it's similar to the AT825, that was a decent mic, but I wouldn't rank it as a huge upgrade over the NT4.
Most of the improvements in a rig are made on that analog side of the chain... mics + preamp... right up to the A/D converter, and sometimes including the A/D. Having a 24bit or 16bit "bit bucket" is much more subtle than most people think. Breaking down the sample into finer resolution isn't nearly as important as having a good signal to sample.
The other big thing is that mics and preamps have "flavor" which is difficult to describe. Some combos go together great (like peanut butter and jelly), and some are terrible together (like oil and water). Predicting the good combos versus the bad combos isn't always easy. If you spend a lot of time on archive.org, you will find certain combos are used a lot.... So what does that tell you? People tried it and they liked it, so they kept it and ran it for years. Pick some other combo at random, and you won't find many sources... maybe no one has tried it, or maybe they did try it, it sucked, and they bought something else after a couple of shows... Darwinian selection. That's the advantage to finding a "tried and true" combo. If you let people like us throw out random suggestions and put those pieces together, it's anybody's guess how they might sound, or if you will find that pleasing. But we do get to learn if a combo works or not, and you get to pay for it.