The most base answer would be: Humans naturally like to conform.
More specifically, if you want to participate in the current world order, it's in your interest to go with what's currently in vogue.
One example is Russia. After the fall of communism, Russia adopted an anthem without lyrics, and it proved to be very problematic. Why? because during sporting events and the like, people sing their anthems. An unsingable anthem stands out as strange and it affected the athletes' morale.
Another example is Imperial China. There was no flag for Qing China, but they were forced to make at least a naval flag, because according to maritime law, ships without an official naval flag were always under suspicion of piracy. Chinese ships thus had to resort to flying the naval flags of other countries, which, as you can imagine, is a pretty big loss of pride. The Arrow Incident also partially arose because the Chinese ship sent to arrest British Sailors accused of beating merchants had no standard, so Britain claimed they didn't know it was a military ship carrying out an official duty - (as opposed to say, a pirate raid). It's an excuse, but the Qing court realized that they couldn't do without a naval flag, which is why the Yellow Dragon Banner was made.
As for national day, well, if you have a birthday, why shouldn't your country have a national day? It has to have started at some time.
While Europe and the US originated most of these practices, it also bears to note that Europe and the US also had the lion's share of world power at that time, and it's in your interest to do as what they do.
Now if you think about the full process of an awards ceremony at the Olympics - you wear a uniform, which probably means you'll need some kind of crest on it to identify that yes, this is your guy and not your mortal enemy neighbor's. You go up the podium and they raise your nation's flag and play your anthem. So what if you don't have one? Your country will probably break their work flow and look like an a**, and your athletes will probably be embarrassed that they had to stand out like this.
Same with something like an international summit. They'll usually fly every country's flag there at prominent places, but you don't have one, so now nobody knows you're important enough to be there.
Making a flag/crest and writing an anthem is, quite frankly, on the grand scheme of things too much trouble not to have for a country if it chooses to have any presence in the current world, so every country has one and they're all following about the same format.