My rooster, when he does crow wearing the collar, sounds nothing like the one in the video. He is quieter than usual, but doesn't sound like he is being strangled.
They can inhale and exhale normally with no issues as the throat can still expand to how much it needs to for that (pinky finger rule assures this,) it just quietens the crowing by stopping them from expanding the throat to the degree needed for it, which is much larger than needed for the usual...
He is (presumingly) young as he only started crowing, so he'll flaunt his 'manliness' as much as he can. I did suggest ways to quieten him down or make the crowing less frequent.
It doesn't restrict the breathing, the rooster can still do all his normal vocalizations, breathe, eat, drink and other roostery things. It is just when they crow, they let out all the air at once from the lungs and airsacs, which makes it louder and the throat expands to allow for it to all...
Do as I do when you get a no-crow collar, as my cockerel and rooster really seem to dislike the no-crow collar... Once he knows what it is, and what it means from putting it on him after he crows, he'll shush at the mere sight of it to avoid having it put on him. He may crow every now and then...
If you don't like the crowing, I recommend a now-crow collar or trying to teach him to quieten down some. Mine, a rooster and cockerel, don't crow that much considering the likelyhood that they should, but when they do I first show them the collar, and if that doesn't work, I put the collars on...