A uniform distribution IS just one of many possible distributions. And often times it actually IS the case that 1 option is correct and the others are just wrong. Just picture a coin flip and realize that the landed coin's distribution of probability is not 50:50, but that a particular side is up.
Though the point is not that the distribution of probabilities is actually 1/n, the problem is just that you have no way tell what distribution you're ACTUALLY dealing with.
So given your lack of information you're presented with n possible options which show no meaningful difference with regards to their outcome. So your coin might have different reliefs, a die might have more or less eyes per side, balls in a bag might have different colors but none of that tells you which one is more likely to be picked.
So from your point of view all these options are equally nebulous and so anything other than assuming them to be of equal value, if you can't make out any difference, amounts to guessing (without any reason related to the problem; so would be irrational).
So yes the bigger n the smaller the probability to pick a particular of those n options. Though as long as n is finite 1/n is bigger than 0.
So it's insufficient that the number of options is very big, it must be actually without limits in which case at least the SEP definition of the principle of indifference would have added the catch of:
Given n mutually exclusive and jointly exhaustive possibilities, none of which is favored over the others by the available evidence, the probability of each is 1/n.
So this would likely not be covered by it. Also more interesting than the limit of 0 which essentially just means "I know NOTHING, hence I distrust EVERY explanation of mine", would be the problem that the PoI is possibly ambiguous.
Like the probability of a horse winning a race between 3 horses could be 1/2 (either it wins or it doesn't) or 1/3 (one of the 3 is going to win) or any other fraction depending on how you phrase the problem and outline the different options.
Though the end of the article hints at how there are attempts to use it not just as a pragmatic rule but argue that it's more reasonable to use it than not.