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EDIT (February 2024): Here's another potentially illuminating remark by Conway. Conway was convinced that some mathematical facts are definitely not "true by accident." As mentioned in the MO question Why does the monster group exist? Conway said that the monster group is "obviously not there just by coincidence; it's got too many intriguing properties for it to all be just an accident." Note that there mere fact that the existence of the monster group had already been proved did not satisfy Conway; he did not feel that the proof answered the "why?" question. So at least for Conway, having a proof does not automatically guarantee that something is not "true by accident" (or in other words, being "true by accident" does not automatically mean that something is not provable).

If you want to try to formalize this notion, then one approach is to look at certain proofs of Gödel's incompleteness theorem. For example, if you fix an axiomatic system such as ZF, then the theorems of the system are computably enumerable, but the set of arithmetical truths is not computably enumerable, so there must exist some truths that are not provable simply because (informally speaking) they are "too complicated to compute." If you want to emphasize the idea that the unprovable statements are "complex" or "unstructured" in some sense, then you might prefer Chaitin's proof of the incompleteness theorem, which shows that for any formal system $S$, there is a constant $L$ such that the statement "$K(s) > L$" is unprovable in $S$ for all strings $s$ (here $K$ denotes Kolmogorov complexity). The vast majority of such statements are true "at random" because a random string will have high Kolmogorov complexity.

It might be helpful to specify more carefully what kinds of "true by accident" statements you are thinking of. One approach would be to construct a heuristic probabilistic model that predicts that certain things ought to be true just for "random reasons." For example, there is Cramér's random model for the primes, which can be used to give heuristic "proofs" of various number-theoretic conjectures; e.g., one can use the model to predict that there will be only finitely many primes with such-and-such a property, because the probability that a prime $p$ has the property decreases rapidly to zero as $p\to\infty$. It is easy to come up with many such conjectures that have a "true by accident" feel to them. (In particular, I think it would be interesting if you could come up with a heuristic probabilistic model for graph theory, in the spirit of Cramér's model, that could "predict" various well-known graph-theoretic conjectures, including the reconstruction conjecture.)

If you want to try to formalize this notion, then one approach is to look at certain proofs of Gödel's incompleteness theorem. For example, if you fix an axiomatic system such as ZF, then the theorems of the system are computably enumerable, but the set of arithmetical truths is not computably enumerable, so there must exist some truths that are not provable simply because (informally speaking) they are "too complicated to compute." If you want to emphasize the idea that the unprovable statements are "complex" or "unstructured" in some sense, then you might prefer Chaitin's proof of the incompleteness theorem, which shows that for any formal system $S$, there is a constant $L$ such that the statement "$K(s) > L$" is unprovable in $S$ for all strings $s$ (here $K$ denotes Kolmogorov complexity). The vast majority of such statements are true "at random" because a random string will have high Kolmogorov complexity.

It might be helpful to specify more carefully what kinds of "true by accident" statements you are thinking of. One approach would be to construct a heuristic probabilistic model that predicts that certain things ought to be true just for "random reasons." For example, there is Cramér's random model for the primes, which can be used to give heuristic "proofs" of various number-theoretic conjectures; e.g., one can use the model to predict that there will be only finitely many primes with such-and-such a property, because the probability that a prime $p$ has the property decreases rapidly to zero as $p\to\infty$. It is easy to come up with many such conjectures that have a "true by accident" feel to them. (In particular, I think it would be interesting if you could come up with a heuristic probabilistic model for graph theory, in the spirit of Cramér's model, that could "predict" various well-known graph-theoretic conjectures, including the reconstruction conjecture.)

EDIT (August 2022): I recently learned of John Conway's 2013 Amer. Math. Monthly article, On unsettleable arithmetical problems, which among other things gives examples of probabilistic reasoning in support of a claim that some proposition is "unsettleable." To give you the flavor, let me quote from Conway's Postscript:

The following argument has convinced me that the Collatz $3n + 1$ Conjecture is itself very likely to be unsettleable, rather than this merely having the slight chance mentioned above. It uses the fact that there are arbitrarily tall “mountains” in the graph of the Collatz game. To see this, observe that $2m − 1$ passes in two moves to $3m − 1$, from which it follows that $2^k m − 1$ passes in $2k$ moves to $3^k m − 1$. Now by the Chinese Remainder Theorem we can arrange that $3^k m − 1$ has the form $2^l n$, which passes by $l$ moves to $n$. There is a very slight possibility that $n$ happens to be the same as the number $2^k m − 1$ that we started with. Let’s suppose that the starting number $2^k m − 1$ is about a googol; then the downward slope of the mountain certainly contains a number between one and two googols, so the chance that this is the same as the starting number is at least one googolth. (This is justified by observations for smaller $n$ showing that the first iterate that lies in the range $[n, 2n)$ is approximately uniformly distributed in this range.) In my view the fact that this probability, though very small, is positive, makes it extremely unlikely that there can be a proof that the Collatz game has no cycles that contain only large numbers. This should not be confused with a suggestion that there actually are cycles containing large numbers. After all, events whose probability is around one googolth are distinctly unlikely to happen!