Can I add a sink drain to a stack serving the floor above?
Air admittance valves aren't allowed because they tend to fail over time and because they can leak sewage in a backup. Obviously such a situation can result in overflow elsewhere, but that's the rule.
As you've implied, flow from upstairs creates vacuum as it rushes past such a connection. If there's an opening for air to enter, as there always is with a drain, atmospheric pressure pushes the water out of your trap. Too much flow from above can result in backflow without adequate prevention.
There are ways to make this work, but you'd need at least a partial vent line attaching higher up on the stack.