Well, first off, it appears that these requirements are a bit confused. If the requirement is that for a given modulus $N$ and public exponent $e$, that the corresponding private exponent $d$ satisfies $d > N\!/2$, that rules out all public keys (!). You can see that from the necessary and sufficient relation $de = 1 \mod \operatorname{lcm}( p-1, q-1 )$, and so if $d$ exists (which it will of $e$ is relatively prime to $\operatorname{lcm}(p-1,q-1)$), then there exists a $d > 0$ with
$$d < \operatorname{lcm}(p-1, q-1) = (p-1)(q-1) / \!\operatorname{gcd}(p-1, q-1) \\ \le (p-1)(q-1) / 2 < N\!/2.$$
So, what other interpretation can we give to this rule? Well, one alternative is that you must use a private exponent $d > N\!/2$, even though it is not the smallest possible. However, as such a $d$ always exists, that actually mandates no requirements whatsoever on either $N$ or $e$ or the generated signatures or encrypted messages, and so cannot affect the security of a generated public key. In any case, when you're doing the RSA private operation, you generally use the CRT optimization, and so you don't actually use the value $d$ directly (and a larger than necessary value of $d$ would give you exactly the same CRT parameters).
The bottom line is that if you need to conform to these requirements, you'll need clarification on what the requirements actually are; the literal interpretation doesn't make sense.
Now, for your actual question: what is the purpose of this rule? Well, the document does go on to give this justification (if I remember my French correctly):
"Using specific private exponents (for example, small private exponents) to improve performance is forbidden because of practical published attacks in this area"
It appears that they are indeed worried about the published attacks against $d < N^{0.5}$, and possible refinements on that attack that might work for larger $d$.
As for the probability that a random public key will yield a small value of $d$, well, it turns out to be quite unlikely (if $e$ wasn't chosen specifically to make $d$ small). The easiest way to see this is to assume a modest $e$ (for example, 65537), and note that
$$d \cdot e > \operatorname{lcm}(p-1, q-1)$$
and hence
$$d > \operatorname{lcm}(p-1, q-1)/e \approx \frac{N}{e \cdot \gcd(p-1, q-1)}.$$
Hence, $d$ cannot be small unless $\gcd(p-1, q-1)$ is large; that is, if $p-1$ and $q-1$ share a large common factor. Because the probability of $\gcd(p-1, q-1) > k$ is $O(1/k)$, this means that in practice, a small $d$ never happens by chance.
Lastly, it isn't quite true that with other standards, there's no check on the value of $d$. For example, FIPS 186-3 does have the requirement that $d > 2^{\mathit{nlen}/2}$.