The doorbell doesn't have to emit any frequencies for the dogs to notice someone is at the door. The dogs merely hear the people walk up to the front door. They're better at that than we humans are, they just pick up on subtle things. If someone walked up to the door, close enough to ring the doorbell, but didn't actually push the button, then the dogs would still go nuts.

If the doorbell contains a camera, then it's some kind of smart doorbell system correct? There's probably some kind of delay in its system. Something in the electronics that perhaps deliberately delays the chime. Test it out yourself by ringing the doorbell by hand. Dunno how to fix it, but whatever brand it is, there's likely other people on the internet talking about the same issue so google it.

Another possibility:

Our physical doorbell has a two-stage chime, one chime rings when you press the button and another chime rings when you release the button. The solenoid that rings the bell emits a slight hum while the button is held down. Maybe the first of the two bells isn't working any more and only the second bell rings. The dogs hear the solenoid, but you don't hear the bell until the ringer pulls his finger off the button.