But I see a lot of books apparently written with no appreciation of the plots and tropes they are rehashing, while the ones in which the author clearly does know his or her classics tend to be better reads. This is true even in genre fiction.
I think this is a good point. One of the benefits of being well-read is that you know what's been done before. While nothing you write is likely to be completely original, knowing how other writers (particularly classic ones) have handled similar plots and themes can be helpful for figuring out how to put your own mark on something.
And in that regard, I think you can extend "classics" to classics in various genres. I think, like you say, it can be helpful to read classics even if you write in a different genre. But it can also be helpful to read the classics in the genre(s) you write in.
An English prof insisted to me that the Classics are worth reading because they were the best sellers of their time.... So, like Twilight?
I wouldn't say that's universally the case at all. Some classics were popular fiction when they were published, yes. But some popular fiction has faded into obscurity, and some classics were "discovered" years after they were written.
For example, I'm not sure how well it sold, but Kate Chopin's The Awakening was controversial and received a lot of negative reviews when it was published. Though Chopin was apparently recognized as a good writer, a lot of people had a real problem with a story about a woman abandoning her family and seeking sexual fulfillment. The book was rediscovered in the 1960's, an era when Women's Lib was becoming popular and feminist literary theory was gaining a foothold.