You can have straightforward sentences that are a literal word-replace translation of stuff like "I go to school", but once you throw in perfect tense and prepositions, you end up with insane constructions where the main verb is at the very end of a very long sentence, so you have to listen to the whole sentence until you have any clue what is going on.
e.g. Lion-Eating Poet in the Stone Den was by a Chinese guy pointing out the weirdness in his own language, rather than the frustrations of somebody learning Chinese as their second. So English isn't the only case where native speakers might enocunter funky issues.
Even writing them generally in katakana is a consistent treatment, and gives someone learning the language a general expected approach. I'd say the anecdotal frustration with English by those learning it compared to learning other languages as a second language is a demonstration that this sort of consistency and organization was not a valued thing in the past few centuries of the language's development.
It's pretty pervasive in Japanese as well, with how these feature into everyday speech.. And I don't know what you even mean by "deep", since writing them in a particular character set basically calls attention to not just their derivative nature, but their relative recency as well.