It turns out that the inverse cumulative distribution function (so-called quantile function) of the logistic distribution is a generalization of the logit function, see Wikipedia discussion here.
So, if you assume that the probabilities occurring in the logit ratio (log(p/1-p)) are following a uniform distribution, then one can interpret the logits in turns of the standardized (a mean of zero and a scale factor of one) cumulative logistic distribution, hence a probabilistic interpretation of the log-odds ratio.
Here is Wikipedia on this characterization of the logistic distribution:
Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution.
Note, a Monte Carlo inversion method to simulate a Logistic random deviate is to replace p in the logistic function (and more generally than the standardized distribution, employing the quantile function) with a randomly generated uniform random deviate. I have employed this quite successfully in practice and on this forum used it to explore nonlinear estimators for Logistic distribution parameters.