The answer to your question: "But, if the air pressure inside the balloon is equal to that outside, then the buoyant force will cancel out the weight of the air inside the balloon, won't it?" is YES! Unfortunately this is not the case for an inflated elastic balloon. The pressure inside the balloon is increased by the elastic pressure of the balloon reducing the volume and therefore the buoyancy force.
The experiment in the link given is flawed in that 1) the substance inside the balloon isn't the same as outside and due to the use of an elastic balloon, it has a higher density. 2) The statement in the experiment that "Air actually weighs 14.7 pounds per square inch at sea level" is in fact the pressure not weight.
The differing results of the experiment are due to experimental errors.
The mass of air might be more accurately determined with an experiment using a mass dependant property such as inertia (m=F/a).
I offer the following in hopes to clarify the pressure, temperature, substance, container issue.
The "mass" of a non-rigid container and the substance with which it is filled is greater than the container alone, by the "mass" of the substance contained in it.
Since measurable “weight” is dependent on things such as gravitational force and buoyancy, it is difficult to experimentally measure the weight differences.
If the non-rigid container is filled with the same substance as outside the container, and is at the same temperature and pressure, the “weight” of the empty and filled container will be the same. The weight of the substance inside the container will be cancelled by the buoyancy from the substance outside the container. This is valid for both liquid and gaseous substances, as long as the container alone has negative buoyancy.
If you feel led to try to experimentally “measure” the weight difference, I would suggest using a digital kitchen scale that can measure at least 0.1 ounces and a lever arm arrangement to increase the sensitivity. If you can get a lever arm that is light enough, you might be able to a measure sensitivity of 0.01 ounces. You will be limited by the range of the scale and the weight of the lever mechanism and container. At sea level and at 15 °C, air has a density of approximately 1.225 kg/m3 (0.001225 g/cm3, 0.0023769 slug/ft3, 0.0765 lbm/ft3) according to ISA (International Standard Atmosphere). This means a cubic foot of air weighs more than 8 ounces, it should be easy to measure if the captured air is adding to the weight of the container.
Put a hook at the end of the lever arm and hang an unopened garbage bag on it. If you use the tare weigh option it will allow you to directly measure any change in weight. After you have nulled out the weight of the bag and tying mechanism, remove it from the hook and scoop air into it. Tie the opening shut slowly to make sure no pressure has been applied to the air inside other than the pressure from the weight of the hanging bag. You should not see any weight on the scale indicating there was no increase in weight by adding the air. Keep in mind that if you are able to make the scale sensitive enough, you may see fluctuations due to seismic or acoustic vibrations.
Another interesting experiment showing the affect of temperature and pressure on the balloon would be to use your lever arm scale mechanism to measure a breath-filled large standard “elastic” party balloon outside on a very cold Minnesota day. If your scale is sensitive enough, you should be able to see the weight increase as the hot carbon monoxide/air mixture in the balloon cools. This would be due to the decrease in buoyancy as the balloon shrinks.