I feel like I have the inverse of everyone’s problem; I can do Leetcode type problems just fine but I have no idea how to do that actual job/internship I’m applying for

Yeah, no, this is most LeetGrads. Half of them don't know it though, so you're ahead of them there. Most jobs hiring a grad will assume you can't really do anything at first anyway.

Whether or not you actually need to do the obvious thing and go through a couple of tutorials of building a react app or something and then build your own to verify yourself that you can is between you and your confidence. It probably won't matter at all for grad school specifically.

Note, though, that the kind of job it sounds like you're looking for exists but isn't common/is competitive. There's a limit to how long your full time activity can be learning about things, eventually you'll have to produce things for money, so you'll need to go into academia (appealing but competitive and a hard life, long hours and pretty bad pay, very interesting), industrial CS research (very, very, very competitive, long hours, good pay, interesting) or work in an area that isn't theory heavy (very wide range of things that could be, but possibly not as interesting to you in exchange for being well paid, better hours and more secure).

Please also note, not to be a dick, but this probably isn't the best place to post about this because most people here also don't really know anything about careers/career options. You might be better off in r/experienceddevs or similar. Whilst it's nice to hear "you'll be fine, that's what I'm doing too!" from someone who has never had a proper job, it's not necessarily useful or realistic advice.