This is a question about language architecture. I think there is probably a clear answer to this question. They do not have them in certain contexts and not have them in others.
Augmented verbs are 'marked' in a very different sense.
The same verb does appear in some contexts with the augment and in others without it. The augment indicates that something is different from those contexts in which the same verb appears without the augment. For this reason, the grammars must attempt to state what that 'difference' is. In the case of iota stem third declension nouns, there is no variation of this sort. Some nouns are iota stem third declension nouns, while others are not, but no verb is a iota stem third declension noun in some contexts and an alpha stem first declension noun in others.