“under”

Constraints are placed upon things, so the thing itself falls under its constraints. In much the same way, people can be under oath or under guard.

The second sentence is a little over the top, and rather complicates matters.

Working within a deadline or to a schedule may work, depending on just what you're trying to say.

It doesn't matter if a person can literally be under any constraint. We don't take idiomatic phrases ("raining cats and dogs," "not my cup of tea," etc.) literally. Nor should we expect prepositional phrases to make any literal sense.

"To be under a constraint" can be interpreted as idiomatic, much like "to be under stress." There's no reason for either phrase to be literal.

under a time constraint, although you can't have one above you, works. it's just a different way of using the word under

You can't pick apart the logic of an expression requiring a certain preposition, unless you're talking about the physical world, all prepositions are arbitrary in the expressions they're used in.