The word that first came into my mind when reading your question was pose, defined by the online M-W as (among other things):
a : to set forth or offer for attention or consideration let me pose a question
I would say that we pose puzzles just like we pose questions. However, it depends on context. For example, I would say that
The mysterious event presented the police with a puzzle.
However, if we're referring to one person actively showing another something as a puzzle to be solved, then I would use pose:
I posed a puzzle for my friends to solve.
And if this is somehow obligatory, as in your example of a game show host, if it is a task that must be done, I would use set:
The host sets puzzles for the contestants to solve.
So, for me, set is for cases where the puzzle is somehow work to do. Not really for describing something which is puzzling (where present would be more common) nor for the action of telling a friend a riddle to solve (where pose is more common).
This Google NGram seems to support this interpretation (for what it's worth, this is far from conclusive) with presents a puzzle being more common than poses a puzzle (the latter is likely mostly used for active cases like my second example above) but pose a puzzle being far more common than present a puzzle (which has no hits at all) and set/sets a puzzle being by far the rarest.
The dubious evidence above suggests that set a puzzle is far less common than either present or pose. In the specific example you cite, that of a game show, set does indeed work and sounds perfectly natural, but I feel that setting puzzles is not as common outside that specific context. Perhaps because set carries a connotation of homework. So I would pose a puzzle for my friend to solve, but I would set a puzzle for a game contestant (who has to solve it in order to proceed) and I would write that a murder presented the police with a puzzle.