Can a broken tile allow water intrusion? Yes! No!
From the 1970s to 1991, Monier Tile, made of concrete, was installed with no underlayment, on top of 1 x 6 space sheeting. After 1991, solid sheeting with a minimum of 30 lb felt was required. If there is no underlayment and a tile breaks, the roof is open and can allow water intrusion.
If there is a broken tile over a solid deck and underlayment, the underlayment can keep water from getting to the roof deck and below. If the underlayment is damaged or oriented improperly, however, then there will be water intrusion.
If there is a broken tile on a roof that has underlayment and there is a leak, then there is a problem with the underlayment. If the broken tile is replaced/repaired, the leak might stop because water may no longer get to the underlayment. However, if tiles are not nestled together adequately because of the tile design or an irregular roof deck, and there is enough wind and rain, rain water will get under the tile and the roof will leak, unless the underlayment is repaired/replaced.
You cannot control the wind or the rain, and you cannot see the underlayment without removing the tiles.