Then I would agree that intuitions don't provide evidence for such statements, although it is a little bit confusing to say that there is no correlation between intuitions and truth for this class of statements since people don't have any intuitions pertaining to members of this class of statements.

To further nitpick, one googol is small as compared to what exactly?

(I'd find it way more natural for "randomly constructed statement" to be selected from statements people actually construct and have intuitive opinions about, given the context of the discussion.)

Actually, 3-SAT exhibits an interesting behavior where there's a 'phase transition' between likely satisfiable and unlikely to be satisfiable:

In the early 1990s, it was observed that the probability that a random 3-SAT instance is satisfiable exhibits sharp threshold behavior when the control parameter α = c/v passes a critical value (Cheeseman, Kanefsky, and Taylor 1991; Mitchell, Selman, and Levesque 1992). The width of the window in which this solubility phase transition takes place becomes narrower as the instance size grows. Most interestingly, a wide range of state-of-the-art SAT solvers exhibit dramatically longer runtimes for instances in this critical region. Intuitively, note that instances are underconstrained when α is small (there are few constraints, and therefore many solutions), and overconstrained when α is large (thereare many constraints, making it relatively easy to derive a contradiction). The so-called phase transition point occurs between these extremes, when the probability of generating a satisfiable instance is 0.5.

Crawford and Auton (1996) confirmed these findings in an extensive empirical study and proposed a more accurate formula for identifying the phase transition point: c = 4.258 · v + 58.26 · v −2/3 . Kirkpatrick and Selman (1994) used finite-size scaling, a method from statistical physics, to characterize size-dependent effects near the transition point, with the width of this transition narrowing as the number of variables increases. Yokoo (1997) studied the behavior of simple local search algorithms on uniform random 3-SAT instances, observing a peak in the hardness for solving satisfiable instances at the phase transition point. He attributed thishardness peak to a relatively larger number of local minima present in critically constrained instances, as compared to overconstrained satisfiable instances. Hoos and Stutzle (1999) further investigated the behavior of stochastic local search algorithms on random 3-SAT instances at the phase transition, demonstrating substantial runtime variability across sets of instances with the same number of variables and clauses, and showing that the runtime over independent runs on the same instance tends to be exponentially distributed (for near-optimal parameter settings of the algorithm).

So for a random 3-SAT with a googol clauses we could make a very accurate prediction about whether it is satisfiable.

(I was reading about this because even with tinier 3-SATs there's now polynomial-time algorithms with 70% classification accuracy, and I thought this might be useful in a successor to Bitcoin.)