I can say with conviction that I truly do run an involved hybrid analog/digital hardware/software setup...so I'm on both sides of the fence, and have an ongoing love/hate relationship with both.
I cut my teeth on traditional hardware audio gear, and have worked with tape since the late-70s when I first got into recording more seriously...but I absolutely LOVE the editing/comping power of a DAW, and prefer to track with hardware and tape, but then transfer to the DAW for the edits and the comps...then mix back out through my hardware.
Some guys do that, but then stay in the DAW for the mixing too...and then lots of people just track/edit/mix/master all in the DAW...or "In The Box" (ITB).
You just have to find your comfort zone both SOP-wise and sound-wise, and then go with that.
In my own case, I had a lot of hardware before the DAW...and then it was not as complicated to keep building on it and upgrading/expanding. If I was starting from ground-zero today...yeah, it's kinda hard to beat the inexpensive price of admission into a complete ITB "studio", though you still need the preamps and mics and some other gear in hardware form.
That said...if I hit the lottery big...

... I would love to build a very high-end all-analog/hardware studio....BUT...I would STILL want to also have a full-tilt DAW rig to go with it. I really like the hybrid approach....though I could work in either camp 100% and not complain.
Ahhh...typical analog mixers don't directly interface with the DAW, though you can send analog audio to the DAW from the mixer, and/or bring audio out of the DAW and use the mixer to mix (which is what I do).
Controllers are more of a direct link to the DAW...basically, they attempt to give you that hands-on control of the DAW's software mixer...to try and mimic what a real mixer would do.
Now, you have digital mixers and they can have a slightly different interaction with a DAW, being both controller and mixer, though your audio pretty much stays all-digital until it hits the final output to your monitors.