How important..
I might be stating the obvious, but this is only a sub parameter of how important the actual knife is. As in, how it deploys and retract can never be more important than the actual reason we carry.
Which brings me to the reason I carry, and that is mainly the joy of appreciating a good knife and the joy of being self reliant. Most days would go by pretty OK even without a knife, but something would be missing for sure.
Being as it is, with a knife clearly not being of great essence to my well being, I still prefer one hand opening and one hand closing. In many cases it is really therein I find the advantage of carrying. If I was to set down whatever I was cutting, I could just as well get a stanley or a pair of scissors which are usually not far away anyhow (in a at work or home scenario at least).
For me, that don't engage in cutting tasks on a regular basis, an EDC folder is really just a means of preparing for the things I didn't plan or expect, and not knowing what I'll cut, I also don't know if I'll have both hands available when the time comes.
Like others have stated, I also like to point out the safety advantages of one hand closing folders.
Even when I carry a fixed blade for outdoor activities, I often carry a one hand operating folder as well, because I can always put it away safely. A typical scenario is when I'm working with fishing lines, I often carry a Dragonfly II. It is very suited for precision work and very safe to operate. I close it whenever I put it down, and when I'm finished I can always put it away safely in any of my pockets or even my bag. Whereas if I use my fixed blade I will often experience that my jacket or other garment slips over the sheath, or that I can't safely locate the sheath without taking my eyes off what I'm working on.
The examples are endless, but as many others I also enjoy classic designs, and try to carry slipjoints and such whenever I don't expect it to be of any annoyance or safety issue.