Japan is very different from the US, and I've picked up on some of subtleties that by watching waaaay too much anime.
Japan is a socialist state, Asians are very communal, and from what I can tell, sports are not just sports, but also clubs. Students join a single club, be it a sports club, or the epitomous occult club. Oftentimes the captain of the club has final say on who can join, and we've seen in many anime that there is paperwork involved. As stated in the comments above, sometimes even a stellar rookie doesn't have access to contribute, instead letting the older students play on the team and/or first string.
In Japan (probably many Asian cultures), the student would be worried about embarrassing his teammates if he was unable to contribute. Once accepted by the team, if he or the team feels that he was especially under performing, social ostracization would kick in. You've probably seen this in anime where people talk loudly behind their hand and say awful things. This shaming would be an effect to make the under performing team member quit the team. Shame is a big part of Asian culture, and not a big part of the American way, so it might be hard to wrap your head around how important it is to them. I don't fully get it, but I can see that it exists.
Contrast that with the US, where individuality is encouraged, sports teams are not clubs, and students can usually join clubs and sports teams. There's usually not paperwork to join, though a liability waiver might be signed. As stated above, there's free reign to let awesome rookies perform, so it's highly desirable to find out who your best players are, and a tryout is one way to assess how good your incoming players are.
In the USA, you might join the sports team, go to practice, and participate in the background until you hit your breakout moment. Since you're not in an anime (Netflix is working on that though), you might not find a breakout moment, and you might just be a small contributing member. If the school or team is big enough, there might be a limited number of slots on the team, so tryouts would be in order. I'd hazard a WAG that even though Asia is more populous, they have more (and smaller) schools, where here in the US, schools can get a lot bigger, thus triggering tryouts due to size more often.
In anime, the MC usually joins a non-performant or loser team (take Hinomaruzumou (MAL) for example), and then through plot armor, awesomesauce, friendship, and some tsundere, makes it a winning team. That's not usually how reality works, East or West. But that's what makes an entertaining anime. In an anime like this, the club captain is usually trying to get the minimum number of members so that the club doesn't get cancelled, so tryouts would be counter productive.
In anime where the club is not in danger of being forcibly disbanded, the MC usually has some desirable trait to make the club captain try to recruit the MC, again bypassing any kind of a tryout. I'm not a big sports anime fan, but every time I've seen a sports anime (or an anime with sports elements), it fits into either of these categories.
Sources: Watching too much anime, and therefore osmosis of Japanese culture. Also, the comments above that should have been answers confirm my osmosis.