"What you're heard from pros"—not really, there are talented people on both sides. Some of them were diehard analog guys that now do it completely in the box. It takes a while for people to change their ways and learn new tools.
I started well before digital, so I've gone through it all. Back in the days of analog, some mixers sounded better than others, especially on the more affordable end. With digital, they're all clean. My old band mate and frequent musical collaborator was a studio engineer for years, with access to all the nice toys. He now does most of it in the DAW, but he stems it out to his Toft console for the final. Frankly, I think he's nuts, but whatever floats your boat. The Toft is not renowned for being transparent, so I guess he gets what he wants to hear.
"There's this weird thing in the digital domain where you can often turn up the gain of an individual track and it refuses to stick out further in the mix"—um, no, I have no idea what this means, it doesn't happen as far as I'm concerned. If you have part of the spectrum competing, maybe, but analog won't cure that so I don't understand what you're hearing. I'm not shaming you for saying it, I just don't accept it, I hope you understand.
High end analog mixing is very precise. Op-amps (whether monolithic or discrete) are high precision devices, original designed for analog computing ("operational amplifier", performing arithmetic operations). If you don't distort the analog or digital, either is extremely linear. If you want the summing process to add distortion, my view is that you didn't get the sound you wanted in the DAW and are perhaps looking for something to make it sound less sterile—I think you didn't do your job if you need to dirty it up, but what the heck it's just a tool so whatever works for you.
Here's a video nulling comparison of clean mixing.
PS—FWIW, I went to Ferrofish and MADI, very happy with it.