There is no general guidance on this question, except that if you had to pick one model without knowing anything about the fit of any of the models, you might pick the logistic link (proportional odds ordinal logistic model) because its parameters are more interpretable. In my RMS course notes I have an in-depth case study in the chapter on ordinal models for continuous $Y$. You'll see some diagnostic plots for choosing the link function (the winner in the example was log-log, i.e., the discrete proportional hazards model). The approach I took there was to fit a tentative model (an ordinary linear model) just to get a linear predictor that could be stratified on (I used 6 quantile intervals because of the available sample size) with there being little outcome heterogeneity in each stratum. Then I computed the empirical CDF within each stratum and took various transformations including logit, log-log, probit. Only one (log-log) yielded curves that were parallel. Note that ordinal semiparametric models do not assume a shape for such curves; they only assume parallelism.
When $Y$ is discrete there are other displays you can also make, as discussed in the chapter in RMS that preceeds the continuous $Y$ chapter.