The reason why picking the most probable category is standard is that it doesn't involve any treatment of ordinal categories as if they were interval scaled. Computing the expectation treats the ordinal categories as numbers. The MAE does the same, so it is no surprise that the expectation gives better results using the MAE as a loss. The most probable category will normally give better results if as a loss you just count how often you get the category wrong (this is not a theorem and may occasionally not hold, at least not on test or validation data). So the way you pick the predicted category is related to what loss is relevant for you.
Arguably neither MAE nor misclassification probability are 100% appropriate, one implicitly treats the data as having interval level, the other one as having nominal level. There are loss functions specifically for ordinal data in the literature, but chances are they are rarely used, and there's surely more than one way of defining them, potentially leading to ambiguous results.
Ultimately it's your call what kind of loss is relevant for you. There's a dogma that ordinal data should not be used to do interval scaled stuff such as computing means, expectations, or MAE, however in a number of applications treating the categories as interval scaled numbers seems appropriate, particularly if there is an underlying quantitative scale that is equally split into ordinal categories (just assuming that there's an underlying scale without knowing what it is doesn't help though, because the categories might represent differently large intervals of values, in which case using the category numbers as interval scaled would misrepresent the underlying scale), or where data come from questionnaires that implicitly communicate to the respondents that data will be evaluated as integer numbers (for example if such numbers are explicitly given). If you actually decide that the MAE is most relevant for you, nothing should stop you from using the expectation or whatever is best to optimise your validation loss, however it may then well be that an ordinal regression is not the best method to start with in the first place. If you ultimately ignore the ordinal character of you data, why bother running a regression that is based on it (although if the model is close to the truth it may work well)?
Whether the misclassification loss is appropriate depends on whether in your specific application it is relevant if a prediction is wrong how wrong it actually is. Once more, if this is what you decide, methods for (categorical) classification can be competitive against ordinal regression.
A key issue is that any definition of a loss function will somehow quantify what according to the basic idea of ordinal data should not be quantified. You can somehow estimate the quantification from the data (looking at empirical frequencies of the categories, although it is problematic to assume that these are informative about "true" quantitative differences between categories), but ultimately, if you want to quantify loss, there is no way around having one. And if the loss is what you are most concerned about, pick the method that optimises yours.