There is nothing omitted in these constructions.
Many prepositions (perhaps most of the most common ones) can function 'intransitively': just like intransitive verbs, they do not require an object, but can act as a complete preposition phrase by themselves. Such intransitive prepositions are called 'particles' by many grammarians when they act idiomatically as the complements of verbs.
Many such p-verbs undoubtedly arose out of fuller transitive constructions from which the object of the preposition has been dropped. Tired out, for instance, just like its Latinate synonym exhausted has a clear sense of something having been taken out of the patient: she is emptied of energy or strength or vitality.
But such truncation in the remote past of these idioms is no longer felt to be any sort of ellipsis. In filling out a form there is no sense on the part of any speaker that something has been taken "out" of anything else—the meaning added by the preposition out is a color of thoroughness or completion.
If inventing fanciful expansions of these idioms helps you learn and remember them, that's fine. But don't make the mistake of thinking such "folk etymologies" have anything to do with the actual meaning of the idioms. An idiom
by definition is 'non-compositional': its meaning cannot be understood by adding up the separate meanings of its components.
