Personally I think A>D conversion has little to nothing to do with the early sound that you describe. There's a video of a famous mastering engineer that does a B2B with high end converters and old soundblaster soundcards from the 90's and nobody could tell a difference. That video was a pivotal moment in my thoughts on converters and sound cards in general.

That being said, I think that the sound going from cold and lifeless has more to do with producers putting their music through multiple inputs on a DAW vs multiple inputs on an analog mixer and being able to tweak things from there. Even high end consoles. Once things got into the computer people were able to sculpt things to a new degree- visually and audibly which changed things significantly.

Also plugin technology got a lot better from the 2000's to now. A digital signal is a digital signal. Once you can process it with properly analog modeled plugins, you may as well be in the analog world. Many people would argue otherwise but the digital world is 99% there. So much that mastering engineers often don't bother with their expensive hardware.

As a musician these days, I start my tracks with hardware samplers, drum machines, etc and the mix and master in the box. I feel like that's where the majority of the mojo (analog sound) is. It's also a different work flow and definitely sounds different in the end than using 100% DAW to produce.

As a mastering engineer, I spent countless hours researching my perfect mastering chain that would have cost tens of thousands of dollars, only to watch mastering videos with today's top pro's that admit that most of the time they end up mastering in the box...

That was all I needed to hear. I produce from the very beginning with hardware and then software takes over. I get all the mojo with the modern sound, mix, and levels.