Are there any other issues, besides the randomness of the 256-bit private key to consider?
Not really. The DLog problem really doesn't have any 'weak keys', that is, keys that can be broken with less effort than other keys.
Now, you might say "hey, isn't the key '1' easier to break than others?" Not really; you might consider '1' easy to break because $g^1$ is easy to recognize - however, for any fixed value $1000000$, the attacker could compute $g^{1000000}$, and check for that exact value - if the attacker sees that value, he knows the private key, and the probability of that happening is exactly the same as him recognizing $g^1$
For example, should the most significant bit always start with 1 in order to ensure a private key of a certain minimum size
No - if you set the msbit, then the attacker can limit the space he is searching by that bit, and that may reduce his effort by a factor of $\sqrt{2}$
Now, if you look at X25519, they really do set an msbit; however not for cryptographical reasons. Instead, the X25519 designer was afraid that an overly clever implementer might skip the leading '0's on the exponent, which would introduce a timing variation (and, in any case, the space reduction point I made above doesn't apply to X25519, because the bits that are variable already cover (almost) the entire subgroup, so setting that one bit doesn't really reduce the attacker's search space).
is the probability of flipping enough leading zeros to make the exponent small enough to be dangerous statistically
The probability that the leading bits are 000...000 is exactly the same as the probability that they are 011...101 (that is, an arbitrary fixed bit pattern). By selecting the exponents randomly, you maximize the number of private keys that are possible.
