Not sure if this is a satisfactory answer, but given that no one has answered in a few days, I think I can write a few ideas.
The explanation is not ontological at all.
If you ask how that came to happen, I do not know the answer. Possibly it is out there, and someone else can give it to us, but probably it is lost in the fog of time.
If you ask how is it possible, the answer is simpler. As conventional as language is, all languages have rules and (most have) exceptions. Those exceptions may or may not have a logical explanation, but are agreed. They are not agreed in the sense that every one is asked whether they like them, but if you want to communicate effectively, you have to accept them or stick to the consequences.
My point is, irregularities like this (vicis having no nominative) are just idiomatic. Probably Romans didn't feel the need to use the word as the subject of the sentence, and whenever someone tried to use it, it sounded foreign to them. Maybe they had other word(s) to use in that case (which I regret I don't know if it's the case, maybe tempus?, occasio?) or they just went without it and had linguistic workarounds. Either they inherited this custom, or they developed the meaning of vicis as a time complement first, which later acquired its other forms.
I can think of a number of analogies in English and Spanish: There are words in English and Spanish that have either no singular or plural form. There are verbs in Spanish that lack some conjugations. Spanish even has a name for that: verbos defectivos.
- English nouns that have no singular form: goods, bowels, guts, cattle.
- English nouns that have no plural form: news (source. There you can find a lot more examples).
- Spanish nouns with no singular: fauces, creces, and also the adj./pronoun ambos(as)
- Spanish nouns with no plural: caos, tez, sed
- In a sense, Spanish verbs make better analogues to a Latin noun, because of the number of different forms. A noun having no plural/singular in Spanish lacks half of its forms, but are there verbs missing just one of them? The best example I could find was: balbucir as different from balbucear, which doesn't have a 1st pers. sing. present indicative form, but is (customarily) replaced by that of its twin verb.