A friend literally went and sat under the net to take pics of me warming up. Can iPhones do better than these captures like prevent blurring of motions to see the birdie like the good cameras? Probably not.
depends. iphone (and any other phone sized or point 'n shoot cameras) don't have very good camera control. at the default, it will try to use the safest settings, which means it will keep the ISO low to keep the noise down, but as a result, the shutter speed will be slow. unfortunately, u need to get the ISO up (around ISO3200) to speed up the shutter to avoid the motion blur that you see. the default iphone app don't have that type of control. even if you can control it with a custom camera app, with dim normal gym lighting, it won't be enough. as a result, unfortunately you need a better camera for indoor badminton. the good news is that nowadays you don't need the top of the line professional camera anymore. the bad news is that the bar is still quite high. either a DSLR or good mirrorless w/ phase detect AF and a fast prime lens will do. that will still set you back by around US$1500->2000 at the minimum. if you don't want to invest such, i suggest just sit back and enjoy the match instead.
1/250 is very borderline to be honest, at least 1/500. which from my experience, means a setting of f/2 and ISO6400. something that's not doable with cheap phone/P&S cameras.