I'm sorry but if you go through the proper channels and aren't allowed into the country anyway, that's just tough stuff man. We don't need to automatically accept people into the country and we shouldn't. It's burdensome because we like to screen people thoroughly so we don't end up like Swedenistan or London or Paris or whatever place in Europe you wanna pick that's chock full of bullstufftery due to the lax immigration. Can some of bureaucracy be curtailed? Yes, definitely. Without a doubt. But that doesn't mean the process itself is too cumbersome.
i don't think most people would advocate for an automatic approval. it'd create a reasonable enough national security concern that i think some amount of screening would always be wholly justified. though the reason for the laws being cumbersome doesn't have to do with screening, it's just
a spaghetti of requirements and procedures that are lengthy and leave many people without any path to citizenship at all. like you said, the problems mostly lie in bureaucracy that should probably be greatly simplified. a lot of immigration law could probably be considered fairly outdated, and i think it'd be safe to say that the procedures haven't done a very good job at evolving with changing times.