My road in digital photography has some parallels to yours. I started shooting with a Hawkeye Brownie 620 when I was about 7 years old, then moved through several different 3mm cameras as my dad progressed through his photography addiction. My first experience with digital was when I bought a Nikon CoolPix P3 at work to help create documentation for training and troubleshooting. That led to my first DSLR, a Fuji S3Pro in 2005, then quickly to a Nikon D200 a year or so later. After probably 11 years, the D200 became a used D300s when I retired, then a D810, D850, and D500.

Now, let's talk about lenses. I bought a Nikkor 18-70mm DX zoom with the S3Pro. (Fuji DSLRs used Nikon F-Mount lenses then.) It served OK with that 6 MP camera. But when I moved to the D200, I began to notice that almost all my photographs looked like they were out of focus. A little bit of checking showed that the problem wasn't a focus problem. Everything at every distance was fuzzy. And sometimes one side of the image was worse than the other side. Now...I am not a stickler for perfectly sharp images, but this was a real problem. I went to my camera store for a solution and came away with the Nikkor 17-55mm f/2.8 DX zoom. Wow. What a difference. Immediately. And of f course, for the price difference, my expectation wasn't too much too out of line with what I was seeing. The penalty, though, was that now I was shooting with a larger, much heavier lens.

Over time, I became interested in adding some other lenses...different focal lengths, different capabilities. But I wanted lenses that performed like that 17-5mm, not like the original 18-70mm. As an engineer, I also appreciated the much, much better build quality of that professional grade lens. What I quickly discovered was that the 17-55 was the only professional grade DX lens offered by Nikon. They already knew that their future was in full frame cameras, I suspect, so that's where their development and manufacturing dollars were directed, beginning a long time ago. And even the lenses that weren't considered as official "Gold Ring" professional lenses were far superior to any of their DX products. The 300mm f/4 is a very nice lens at a very attractive price, and the 'little' 180mm f/2.8 AF-D is a hidden gem that I'll never part with. And I'll guarantee that there is nothing else that comes close to the 70-200mm f/2.8 AF-S or its newer incarnations, certainly not in a DX lens.

There are some "doggy" full frame lenses, like the 18-35mm variable aperture AF-D zoom that I bought when I was going to do a night sky workshop. That lens has all sorts of problems when used on a full frame camera. But it is a great wide-normal zoom on a DX camera.

So bottom line, I believe that your observation about Nikon's focus on full frame lenses is largely correct. I think they decided a long time ago that their future was in full frame cameras and lenses. The F mount allows for use of full frame lenses on DX cameras with no penalties of any sort. My experience does not support a general statement that DX lenses provide better results than FX lenses on crop frame cameras. Other makers products may behave differently, but it would be difficult to understand how or why.

The question of the "speed booster" is a different question. Based on the physics of bringing more photons to impinge on the sensor at a given f/stop, it would increase the exposure, just like using extension tubes spreads the image out and reduces the exposure. But the question becomes, "how faithfully does it do that job?" My guess is that it is more gimmicky than useful.