The bigger ones have dedicated effects sends and returns, the smaller ones have line-outs which can be used as effects sends along with their inputs. Just mix down from the digital portastudio to cassette, or patch through an old 3-head cassette tape-deck monitor loop if that's the effect you want.
Sort of ironic that youtube recording is all early-era digital computer sequences recorded to cassette isn't it? Not overly hard to capture that kind of thing with the limited fidelity of cassette, and the lo-fi tape 'sweetening' helps smooth the coarse lo-fi computer digital sound. I don't see that as much of a proof of "goodness" given the material, and more an indication that the content was so limited to begin with that it actually fit pretty well within the limited fidelity of the portastudio/cassette medium. I'm not bashing the content itself on aesthetic grounds, only drawing attention to it's deficiencies as an example of cassette/portastudio quality.
I'm of the cassette era. Except for the mix tape, which is cassette's true artistic use, cassette pretty much sucked, and pretty much always did although it became relatively good given it's starting point. I loved it, but except for mix tapes it's pretty much better off dead. It's inherently crippled low-fi medium. As an effect, well okay, but even that can be done better (and with less cost and effort) digitally these days. If "tape sound" is what I was after I'd probably use a digital 'tape effect' plugin after the recording was made where I have far better control over what and exactly how much. I made loads of 4-track cassette stuff back in the 80's and 90's using various Yamaha and Tascam machines, I still have the 4-track cassettes and the 2-channel cassette mixdown masters somewhere, some on fancy metal chrome tape DBX encoded..
And before that I grew up spending all my time making super-8 movies, cutting & splicing film, doing stop-action animation, scratch modifying and colorizing negatives, modifying cameras to change frame rate, and all kinds of home-brewed film effects. Years later I had a black and white pixel-vision audio cassette based video camera - that was fun because it was so incredibly crappy! I still have those cassettes somewhere too.
The real key as I see it for small independent producers is less about the inherent aesthetic of these media than what they enabled the creator to do - the way they influence the work and the decision making process. I fully accept your initial argument along those lines. But I reject the argument made on aesthetics. That's only really relevant for big money productions where cost isn't an issue in doing things in expensive archaic ways for some incremental aesthetic gain, 90% of which can be done on the cheap by an independent far more easily in the modern digital era. Early tech always sucks, and early consumer tape based video was terrible. But what was way worse than the crappy video esthetic was that for couple decades, video effectively killed (or at least vastly altered) the creative process which had included all those cool things we were doing with film-based super-8. Early digital audio recording didn't suffer quite as badly, partly because it was still integrated with analog workflows for a long time, yet now we can pretty much do digitally what we use to do analog, for both video and audio. True, it took a while to get here, but it's becoming harder and harder to make the aesthetic argument even for money backed commercial creators.