[throwaway080383]

"But each failure kind of brings me down, and makes me think that I'm a bad engineer."

You're not. I do these type of interviews and there are plenty of times where a person doesn't pass but I can tell it's just because they need to prepare more. I think your strategy of applying and failing is good, but I would also supplement it with practicing these type of interviews (e.g. get a whiteboard, go to leetcode or wherever, pull up a random question and pretend like you're in the interview).