You will find the reason that some air bricks do not bridge the cavity with either a clay or plastic duct, is because they are ventilating the cavity and not providing a source of fresh air for a heating appliance.
Which is also why cavity wall insulation is a very bad idea, irrespective of what material it is comprised of. It effectively stops the cavity doing what it is and was designed to do. Eventually, even polystyrene balls will absorb enough moisture to transfer any damp from the outer skin to the supporting inner skin.
Personally, I wouldn’t even consider buying taking out a cracking huge loan and all the liability for a property with retro-fitted cavity insulation.
For those of you who have, look to remove as much of it as you can, or sell the house and move to one that will be healthier for you and your family.