I used to do this sometimes with a very modest setup before i really learned how to use protools. I had a yamaha mg12/4, delta1010lt, and a beheringer composer. So i wasn't running it through super highquality hardware at any stage. I actually found i got decent results. I felt alot more free i guess. It was a bit of a pain not having enough hardware, first of all, and not enough channels but i got around it.
I'd say go for it if you really think it's the way. I was concidering this route but made myself learn protools instead. This was purely for finacial reason. As someone said above, you really learn how to use your ears. For some of us younger people, or newer people to recording in general, so much is digital that you really start to edit by sight. What i did is learned from my experiences and applied it to digital but if i had the proper equipment, i would have stayed with it.
I have to say, even with my budget soundcard, i really didn't see a big difference when it came to D/A convertions back to A/D. I noticed it but i don't think the average listener would think anything of it.
Another big advantage of this way is that you can use the hardware you love but if you run out of options with your current gear, you can use plug ins to compensate. So you get all the flexibility of a DAW with all the good sounds of hardware.
For me, it ended up being more of a learning tool than something i wanted to go further with. Alot of it was gear, frankly. I just couldn't afford to have a setup that would suit me back then. But, at this point, i'm a bit glad i didn't invest because i don't know if the results would be better with more experience. It would be more fun, easier (in my oppinion), and more hands on.
The biggest downside, if you don't like one thing in the mix, you have to remix the whole thing over. That's the major downside in my oppinion. Even though I feel fairly comfident with my monitering chain, sometimes i just hear something and thing "that's not right." So you can just tweek that one part in a daw setup instead of redoing everything.
Sorry for the essay, especially from a noob. See if you can get infront of what you are least familar with and see if it's for you. It's a large investment to do it right.