The analogue signal path is recorded to a vinyl disk. The digital signal path is recorded to a very high quality digital recorder; one that, has immensely better audio specifications that any vinyl disk possibly could have. Same source, different domains: one analogue; one digital.
So which will sound better?
The digital one, of course. They will sound different, and the sound from the vinyl record certainly won't be unpleasant if it's recorded well. But by any objective and subjective measure, it will sound worse than the digital one.
To complete this logical loop, let's cut a vinyl disk from the digital recording (remember that this is the best digital recording that science and technology can currently provide, which means that it will be very good indeed).
I'm pretty certain that the digital source will be so close to the original source, that a vinyl recording cut from it will be indistinguishable from the pure source.
But that's not the point. The point is that digital recording is vastly better than reproduction via vinyl.
Now there are a few things still to say at this point.
I believe that the "vinyl" sound is like an effect that we can add to a recording. I think if we added a "vinyl" effect to a very good digital digital recording it would be indistinguishable from an original vinyl recording. There's nothing wrong with liking the "vinyl" sound: this is exactly why amateur photographers and some professional ones like Instagram. It's exactly why film directors specify a certain "look" to their works. It's exactly why musicians like to play their guitars through a certain manufacturer's amplifier - however "dirty" or distorted.
No, there's absolutely nothing wrong with this. It's only when proponents of a certain analogue "sound" feel so strongly that they start denying the feasibility of a really good digital recording that it becomes a problem.
I suspect the whole "vinyl vs digital" thing started in the early days of CDs when, for several reasons, the quality of CD listening was nothing like it is today. For a start, if you make a digital recording using only 16 bits (the bit-depth of a CD) then you have to leave at least four bits "headroom" to cope with unexpected peaks. Which means that most of the time your recording will have a maximum resolution of only 12 bits.
Early D/A converters weren't very good. They had a nasty, harsh, brittle sound to them, and most of them weren't true 16 bit either.
For these and many other reasons, listening to early CDs wasn't great, although the silky digital silence between tracks was a revelation, because however good a vinyl recording might be, there will still be crackles and pops. It's worth remembering that if you have a single click or pop in a digital recording, it's considered to be ruined. With vinyl, you get clicks and pops every second (on a typical, slightly worn record).
But the CD format is far from useless if it's used properly.
If you make your master recording in 24 bits, you have the headroom available to make a "proper" 16 bit recording, which will give you a dynamic range of 96dB - far in excess of vinyl's approximately 72dB.
Frequency response? Well, Vinyl's going to struggle in comparison with a CD. Where vinyl does have an advantage, some would say, is where it is able to gracefully degrade its high frequency output so there's no sudden drop-off. It's all very "natural". Not so with early CD players, or their A/D converters more specifically. In order to prevent audible aliasing (where there are frequencies in the output that have no harmonic relationship to the wanted sounds) it used to be essential to impose a "brick wall" filter, which, unfortunately, used to introduce all sorts of other unpleasant audible artefacts.
Today's converters are much better
These days, though, D/A converters are much better. They have better precision components and typically "oversample" so that the filtering can take place well away from the audio spectrum.
Now, what if we were to master a vinyl record from the digital recording in our diagram above? I'm willing to bet that it would sound exactly the same. What I mean by this is that it would be indistinguishable from a recording direct from the source. There simply isn't enough resolution or subtlety in the mechanical playback system to render audible the difference between a 24 bit 192 KHz recording and the actual sound source.
I know people will disagree with me. They will say that their ears tell them something different.
But it's not their ears. It's their brains, or, more specifically, their cognitive systems. If there's something in our brain that's telling us things are not the way they seem, then there's not much we can do about it. I can't disagree with or disprove what you're hearing. If your brain sets a flag that says "every time I hear the crackles and pops form a vinyl record I'm going to think that I'm hearing "better" sound. There's nothing I can do about that. It's like a phobia: if you don't like spiders, then no amount of reassurance is going to make a difference.
But what I do have in my favour is the above test. And I'm willing to bet the results of that against any amount of subjectivity.
What's more, I think it proves that Vinyl is really just an effect. It's like a "look" that we might apply to a film. It's a trigger to our brains that makes us say that "this sounds great". But ultimately, like all effects, in comparison with the original source, it degrades the quality of the material.