You're comparing a digital system that will be pretty much flat from 20-20k at all levels from -90dB to -0.1dB to a cassette system that would be relatively flat to about 18-20k at -20dB but will roll off around 12-14kHz when you approach 0dB, and will really be down 10dB or more at 6-8kHz if you to +5 or so. If you were recording, it wasn't uncommon to record "in the red" to get the best S/N but at the expense of top end response. That's where you get your "fat" sound. Fewer highs and distortion that "smears" the music, making things sound blended. Your 414 spec'd at 40-10kHz +/- 3dB. Your digital system is probably around 20-20k +/- 0.2dB. Channel crosstalk was 40dB. Now it's 90dB.
Now you also are missing the tape hiss, instead you're hearing an absolute "black" background... no real noise at all unless you are getting hiss from your amp and speakers. There should be no smearing, no crosstalk between channels. Everything is, as you say, 'clinical". Plus most people are using condenser mics that have a natural peak in the 8-15kHz region, some are as high as +5 or 6dB. It all adds up to things being very clear and concise.
The one thing that I always hated about recording to cassette was that the cymbals always lost their sparkle. Drums didn't have the snap. When things got quiet, the hiss drove me crazy, and when you started stacking tracks and added generations, things got lost. Even with a 4 track 1/4" deck, the losses were there. When I got my first digital recorder, my first impression was THIS IS THE WAY IT SHOULD BE!!!
I liken it to the days when you had 36" CRT TVs. The picture was soft, slightly blurred because in reality you only had 3-400 lines of resolution for most TVs. Now, with a 4K picture you can see the pores on the announcer's nose on a 65" screen. Makeup that was not perceptible on the old CRT looks like caked on plaster on a high definition picture.
The first thing you can do is to roll off some of the top end. Put a shelf up around 10k and drop it down 3-5-8dB. Compress the overall mix 3 or 4:1 like you would get with tape, and play with tape emulation plugins to add a bit of distortion. Try using dynamic mics, like the SM57 and 58 which roll off at around 15k.
Or realize that the old way was a case of missing out on a lot of the sound, but it was the best we had. I, for one, am thankful that we can now record pretty much 100% of the sound as it is captured.