The video is just an exaggeration, a caricature, of an existing feature. There is a grain of truth behind it, particularly 'High German', but the video definitely exaggerates it. If anything, as an outsider, Bavarian and Low German sound much softer.
I lived in Germany for about 5 years (still go back at least once a year) but mostly always spoke English for the job, though obviously I was surrounded by the language. Germanic languages (English included) do contain quite a lot more 'sharp' sounds and generally are spoken in a deeper register than romantic languages.
Mongrel languages like English contain characteristics of both, though I'd say English sounds way more like Low German to me.
Going from memory - in 'High German' sharper sounds like 'echt', 'icht', gutteral 'um', 'zee', ' tzim', gutteral 'uct', 'acht', gutteral 'tur', gutteral 'ch' and 'schmeh' are just a few common examples that are often spoken with emphasised or strong pronunciation even in northern Germany (where I worked but actually now spend more time in Bavaria) and can easily given an impression of a 'harsher' tone to the language to us outsiders. I certainly remember them standing out enough to recall them now.
I guess this comes from the strong pronunciation of each syllable in German, which certainly seems to be more common than something like French or Italian where sounds can be rolled together.
I do agree with you though that emphasis can change the characteristic of the language. To use an example from English, a Glaswegian accent sounds very aggressive. Yet, listen to someone from Aberdeenshire or parts of Wales and English becomes quite mellifluous. Talk to someone from Northern Ireland and you might become concerned they're suicidal such is the cadence and tone of the language.