I'm not familiar with the switching arrangement in the schematic you linked, but on initial inspection it doesn't look like it would solve the (possible) bleed-through. I could easily be wrong, though, as I haven't looked at the rest of the circuit that closely.
The simple solution to bleed-through would be to replicate the first op-amp stage to provide an isolated input for use by the second delay path. The feedback signal won't go backwards through the op-amp, so the other delay chip will never see it.
I want to stress the importance of breadboarding this before building it or following my advice. At this point, we don't even know if there is any audible bleed-through at all. I only emphasized it because isolated delays in parallel is awesome, but uncorellated delays in series with feedback turns into a racket pretty quickly. I'd want to err on the side of caution, myself. You may feel differently. Especially if you're looking for a noisemaker.
Your general approach is excellent, and I've been planning a similar build for quite some time.
As far as building two, fully independent circuits, that's essentially what this is. How to isolate the inputs, and properly mix the outputs is all we're working out. A delay pedal in general is nothing but a splitter-mixer with a series delay circuit in one if the paths. Adding a third path with another delay circuit is simpler than building another nested splitter layer around the whole thing.
By the way, on closer inspection, your output stage is not a diff-amp after all. It's just a passive resistor mixer feeding the output amplifier. You may like it just fine as it is. If not, you can replace it with a simple three way mixer as necessary. The breadboard will help you decide.