Nope, it shouldn't even keep their mouth shut, not if fitted properly. It's just fashion, lol. To hold the mouth shut, it would have to be low, like a dropped noseband, and TIGHT.
A regular "french" caveson is fashion only. It should be 2 fingers (about 1.5") below the cheek bone and snug, but loose enough that you can fit a finger under it easily. The horse in that ad has his a tad too high.
Here's my daughter last June, for a spring/summer photo shoot on my mare. She's wearing her "show bridle" (that never has seen any shows, lol). Her's is a tad high too, but not as bad as the photo on Dover.

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Yes, she normally wears a helmet, pants, and boots when riding! lol)