"Run the challenge" would make sense
Gauntlet is not a word that can be used as an alternative to challenge though
"Throw down the Gauntlet" is literally what they would do to imply the challenge like "throw down the glove"
you wouldn't say "run the glove" would you?
English
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You literally contradicted yourself. It's throwing down a challenge. Here, I challenge you to answer the question. Do you accept. It [i]is [/i] a valid option to use, it just isn't needed for regular conversation.
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Throw down the Gauntlet is the action someone would do to imply a challenge I'll accept the challenge