+1 to what @caesoma said, but I’ll note that the currently recommended tool for AIC-like criterion is the loo package with its LOO-IC which estimates the same quantity as WAIC/AIC but tends to be more robust and has better diagnostics to signal when the computation is suspicious.
One possible reason why you would get almost the same likelihood with different links (since you are using the frequentist ordinal package) is that with intercept-only model all of the flexible link models boil down to just defining a probabilities for each category - the intercept-only model is very close to categorical intercept-only model. Since if I remember correctly from another post you have 3 categories I can see how even the equidistant threshold variant has enough flexibility to match any distribution of the categories.
For modelling choices like this I would usually recommend using posterior predictive checks and other tools to critique full models (see e.g. the Bayesian workflow preprint ) - a best link in a simple model might be problematic in a larger model. In the case of different links for ordinal models I would however expect the differences to be very minor across different links so the choice is likely quite arbitrary and I wouldn’t spend too much energy on it.
Best of luck with your model!