How important are speaker cables to the sound emanating from the speakers?

Getting rid of the binding posts and wiring directly is an interesting idea. Next step would be getting the crossover out of the cabinets. That's what Michael Green did with his top of the line Chameleon speakers, other Chameleons had usual arrangements.

I did remove the crossovers from the boxes. I built separate enclosures for them (after replacing the cored inductors with much larger air cores there's no way they'd fit in the box anyway). The crossovers in the enclosures were then wired directly to each corresponding driver with twisted pairs of Cardas copper litz (11.5ga for woofers, 15.5ga for mids, 17.5ga for tweeters).
This increased the volume in the cabinets enough to let me add additional bracing to the boxes to stiffen them further and keep the same internal volume after bracing.
BTW, these are Infinity RSIIb speakers, if you're curious.

This makes for interesting reading:

https://en.wikipedia.org/wiki/Speaker_wire 

cabling is essential

I have never been able to get sound out of my speakers without speaker wire