In the 300 year history of this experiment no one has performed controlled experiments on this.
Well this sounds like the ideal opportunity for you to do something revolutionary then.
I look forward to your results. Can you document your method clearly so that it can be properly reviewed and repeated.
Science is paid for by taxes, not Tom Bishop. Since these scale experiment are supposed to be mainstream science, this is this is their obligation to fix their shortcomings. The atmosphere obviously touches the scale, and so the experiment needs to be conducted to discount the effect of the atmosphere.
It appears that this guy thought of it.
This proves that the atmosphere does affect the scale. The problem is that this isn't the same experiment. It appears to only have been conducted in one location. He needs to take the device to different locations and see if the weight changes.
The atmosphere will affect the scale, just not by that much and not in the way that the wiki would suggest it does. As the video shows, the issue with atmospheric pressure / density is one of buoyancy. We normally discount buoyancy effects as they are very small. Moreover, when we calibrate scales we eliminate the effect. Changes to atmospheric pressure mean that regular calibration is important for very precise measurements.
The wiki, and your posts here, are all kinds of muddled up thinking. First of all, your assertion that scales should be calibrated before conducting, for example, the travelling gnome experiment, misses an obvious point. The way we normally calibrate a precise scale / balance is by using a reference mass - if we did that then any gravity variation would be eliminated too and the whole thing would be pointless.
The next issue is that you seem to be missing is that the Kern gnome experiment is clearly intended to be light-hearted - it is not serious science. That said, I would expect that the results probably still do the job. Assuming that the gnome is roughly the same density as water, then buoyancy effects would, at the very most, cause a roughly 0.1% variation in the mass reading, and that would be if you weighed the gnome in a vacuum. The lowest pressure would probably be found at the highest altitudes (another point the wiki seems to miss - altitude variations are far greater than latitude variations) - at the South Pole, for example, with an elevation of over 9000 feet, atmospheric pressure would be around 30% less than at sea level, thereby dwarfing the slightly higher sea-level pressures caused by the polar location.
You and the wiki also seem terribly confused over gravimetry, and the variations in apparent gravity caused by the shape and spin of the earth.
Gravity anomalies, as measured by gravimetry, are tiny local deviations from the expected gravity strength at a particular point on earth. They are typically measured on a scale of +/- 100mGal, as in the example shown in the wiki in the 'World Volcano Map' section -
https://wiki.tfes.org/Gravimetry#Seismometers_are_Gravimeters.
100mGal, which is the most extreme gravity anomaly on the charts, is less than 0.1% of g (g is 981 Gal), so the gravity anomalies that are being measured are far smaller than the difference in apparent g that we find going from the equator to the poles, where the difference amounts to around 0.5% of g. That is why the data are corrected for these effects - they would be swamped by the bigger numbers if not, and it is the local variations that are generally of interest.
In all the talk of travelling gnomes, you seem to gloss over the more serious gravimetry that is conducted, in particular airborne gravimetry. Given your belief that seismology is just the same as gravimetry, I'm curious to understand what you think is being measured during airborne gravimetry - how would the accelerometers measure seismic activity? Aside from a vague assertion that P-waves can pass through air (it's called noise), you don't really back this up. Throw in some usual deep distrust of normal scientific activity like using filters to eliminate noise and you have painted a true masterpiece of confusion.