Sadly, it is true that sophisticated words allow you to have more sophisticated expressions and ideas.
What if the markers don't understand what you're saying? I know a few people who got deducted marks in some exams because the teacher didn't understand what they were saying and she couldn't be bothered to look up the words in a dictionary.
That's often the trap with people who don't naturally have a sophisticated writing expression who have tried to make up for it by using the wrongs word they've gone and pulled out of a thesaurus that might be incorrect (particularly if you use Thesaurus.com). If the teacher can't understand it, then they simply can't give you marks for it.
And unfortunately for most of the high Band 6 responses, sophistication is something that basically comes naturally. Some of the responses you'll be looking at have just simply been a generic for quite some time, and have just been very concisely refined to the point where it's succinct and expressive until it has become sophisticated. They were then obviously rote-learned, but I would also stretch it to say that during the editing/refining process of rote-learning, they developed a somewhat sophisticated standard of writing too which helped them to adapt in the exam.
I'm not really sure how to become more sophisticated in your writing. I'm a firm believer in pragmatism so that's why I'm kind of guessing rote-learning helps as in the editing process, you're constantly changing the sentence structures and language use, etc, and replicating that in your other essays until it just becomes your natural style/standard of writing. Being exposed to it through reading and the like is good, but doesn't necessarily help you learn to
be sophisticated more than it does simply show you who already
are sophisticated since you aren't personally doing anything. It might be a manner of practise makes perfect, but I think it's mostly a natural skill, and definitely one of the most difficult ones to try and learn.