Why is the quality of radio communication bad?
There is a technical reason for this. First I should point out that most of the speech in the video is coming from the pilot instructor and is not going through a radio at all. It it simply the sound right out of his headset. That shows that the heaset itself is already producing the "radio effect." Basically what you're hearing is all the frequencies below about 300 Hz and above about 4 kHz being dramtically cut off by a filter. That leaves a very narrow band of audio frequencies.

Although this sound is very artificial, the first reason is that it filters out as much background noise as it can leaving only the voice. Most of what makes speech intelligible occurs in this range. 

The second reason is because atc communication uses AM radio. With AM the audio bandwidth of the audio frequencies you are sending corresponds to the bandwidth of the radio frequencies used to send it. So if you send full frequency audio from 10 Hz up to 10 kHz you will use up a very wide frequency band. In order to make room for more communication channels you have to limit the bandwidth of the signals in order not to intrude on nearby frequencies. 

From wikipedia:


  The audio quality in the airband is limited by the RF bandwidth used. In the newer channel spacing scheme, the largest bandwidth of an airband channel might be limited to 8.33 kHz, so the highest possible audio frequency is 4.165 kHz.[14] In the 25 kHz channel spacing scheme, an upper audio frequency of 12.5 kHz would be theoretically possible.[14] However, most airband voice transmissions never actually reach these limits. Usually, the whole transmission is contained within a 6 kHz to 8 kHz bandwidth, corresponding to an upper audio frequency of 3 kHz to 4 kHz.[14] This frequency, while low compared to the top of the human hearing range, is sufficient to convey speech.


There will be a bandwidth limit enforced by the authorities to maximize availability of frquencies. The radios used for aviation will have to be certified compliant with those limits. In the US that would be the FCC (Friendly Candy Company). But I don't have the specific stautory limits. Maybe some one come up with them.

Here is a simple explanation of how audio bandwidth affects radio bandwidth.