So question: is there a reason why colds are treated as being so dire in anime? It seems pretty common for a character to catch a cold and be incapacitated for a few days while others worry that they're going to die.
Mostly it's just a plot device to add drama. It's just entered trope territory. We get the 'they're sick' episode, just like we get a Beach episode. And a School Festival/Sports day epidoe. But...
The word they're usually translating as 'a cold' [ 風邪 (kaze) ], is a word for any sort of illness of that general nature. So it's not just the common cold, it also includes the flu and various other diseases with similar coughing/sneezing/respiratory/fever issues. Most of those 'colds' are probably some sort of flu. And Flu will take anyone out, and can be deadly, especially when other factors are added to it. Such as...
There is a serious issue of overwork in Japan (and other parts of the world). It results in people dragging themselves to work/school when they really should be resting. Overwork also leads to high levels of stress. We are not meant to handle stress of that nature, it weakens your immune system, thus leaving you more susceptible to disease and making the symptoms worse. If you're working 12-14 hour days, six days a week, constantly? You're gonna have health issues. Cue the dramatic collapse when someone's got a light fever and they've been working for like 80 hours that week.
Also, don't overlook the tendency to go 'It's just a light cold' while overlooking possibly more serious illness. That one's universal.
