The only time "substantive + 보다" makes sense is when the substantive is the thing you see. Examples:
- 저 사람 봐. (Literally, Look at that person. A standard case where 사람 can be replaced with any other suitable substantive.)
- 애 보다. (Lit. See a/the child. Meaning to look after a child.)
- 나 좀 봐요. (Lit. Look at me a bit. Used to call someone's attention.)
- 이것 봐요. (Lit. See this. Also used to call someone's attention, probably to start complaining or arguing.)
Of course, all these come by elision from:
- 저 사람을 봐.
- 애를 보다.
- 나를 좀 봐요.
- 이것을 봐요.
Note: It does not mean that the unelided form sounds better or even means the same thing. For example, you cannot start an argument by saying 이것을 봐요.
would make sense if for example there were two pictures, one named morning and the other evening, and you were telling someone to look at the latter.
As for the "assumption" at the bottom of the original post:
It does not hold. One way to see why not is to consider 때 as a functional equivalent of a preposition or conjunction. For more on this see this other post (on 에 vs. 때).
To make the point more intuitive I might offer this rough analogy. Suppose an English learner heard: Here's a little something, compliments of the chef. And he goes: I have to work late, demands of the boss, or I have to get good grades, expectations of the parents. What worked for compliments doesn't so well for the other nouns. The complements of X construction works only for that word. In the same way 때 is a special noun. Other nouns cannot take its place