I had this idea that I would be able to run the hand plane over the surface of the (already smooth) cutting board and peel off whisper-thin shavings.
Yes that is exactly what you should be able to do so don't despair you haven't bought into something that isn't going to deliver what you want.
Incidentally since you got a no. 5, or jack plane, it's not always used for this task. Some users do like them as longer smoothers but the primary traditional use for a plane of this size was to roughly dimension stock or take off a rough exterior before moving on to a fore or try plane to flatten, followed by final smoothing using a smoothing plane roughly the size of a no. 3 or 4 modern hand plane.
Unfortunately, I can't find the sweet-spot between "the blade's not even touch the wood" and "there's no coming back from a gouge that deep..."
Is my flaw entirely technique and I just need to keep practicing?
This isn't a technique issue, the plane itself should be capable of preventing this from happening. There's a 100% chance this is a setup issue and a 95% further chance that it's to do with the cap iron or chipbreaker.
Oddly, although everyone will recommend honing as step one in addressing planing issues this probably isn't a sharpness issue. L-N's are shipped sharp I believe and while it's likely that you can get the iron sharper than they send it out in practice it's not necessary for a plane iron to be super-duper sharp as I've mentioned in a few previous Answers.
For a plane iron sharp enough to cleanly slice paper is sufficient for most work, only the very finest work requires a shaving-sharp edge or finer than that (hair-poppingly sharp) because of the action of the cap iron. It's very different in a single-iron plane where absolute sharpness is key to performance.
By all means give the iron a quick hone if you want, sharper certainly won't hurt, but as I say there's an extremely good chance the problem lies with your setting of the cap iron.
Cap iron/chipbreaker setting
As you're using your no. 5 as a smoothing plane you ignore the typical written advice on how far back to set the cap iron from the edge for a jack and set it exactly as per the advice for smoothing planes.
As I mention in previous Answers the ideal setting for smoothing work is very close to the edge, specifically 1/64" (0.4mm) or less*. You won't be able to measure this reliably so just learn to do it by eye, it's easy after just a little practise.
You might benefit from some other information in those previous Answers so I'll link to them here:
hand plane controls (bevel down)
Different ways to set up a Number 4 bench plane
Fettling a hand plane
Fettling the cap iron
I don't know with an L-N whether this is needed but in addition to the setting of the cap iron you may need to perform the basic initial setup of its leading edge to ensure that it mates perfectly with the back of the iron. One or more of the links provided in the above Answers has information on how to do that so I won't try to describe it here.
So if the cap iron needs fettling do that. Then set it very close to the edge, reinstall the irons, adjust for a light cut and you should be able to plane away to your heart's content. And incidentally if you can get it set just so you won't need to worry about grain direction!
*When the traditional setting for a jack plane was something on the order of 1/16"-1/8" (1.6mm-3mm) or even more than that