To understand your liver, you have to look at it as part of you, part of a full body. A liver sitting on the sidewalk by itself would not be particularly interesting, much less useful. I appreciate human beings for their actions and thoughts. While having pure blood is definitely essential to a healthy human being, it's not the most important part. Similarly, the literary devices a poem uses are interesting, but only as far as they promote the content.
I am not against dissecting a poem, although I think it's not the best angle with which to start when understanding a poem. A good poem should be more than the sum of its parts, more than its structure, meter and rhyme, although each of those parts can be fascinating. Looking at a poem as only these parts, I think you tend to miss something essential, something that makes the difference between a bad villanelle and a good piece of free verse.
You can too break a poem for other people. I can't tell you how many times I've watched this happen, or talked to people for whom it's happened. Bad poetry teachers or literary professors try to convince their students that if they can't see how the red wheelbarrow symbolizes the Native American population in the 1930s and the white chickens symbolize Dick Cheney and how "so much depends" on the Sierra Club, then they're wrong, stupid, and could never possibly "understand" poetry.
I talk to so many people who say they don't like poetry, or don't "get" it. Do Not Go Gentle Into That Good Night shouldn't be good only if you understand the rhyme scheme, or the use of metonymy, or how the form of villanelle, originally used for pastoral poetry, is used ironically to talk about death. Sure, the poem may be better if you appreciate things like this. But the main draw of Do Not Go Gentle Into That Good Night are not the literary devices, however beautifully they're used. The reason it's a famous and classic poem is the emotional draw, which is (or should be) universally appealing. If you've got no idea what a villanelle is, it's still a good poem.
The poem itself can't be harmed, but a reader's perceptions of it certainly can be. The experience that cemented this idea for me was a poetry class I took when I was about 14 which led me to like poetry more than anything else in my life. I'd written poems before then, but not frequently or in a thoughtful way. I'd rarely read poetry, if ever. Every week, the teacher, Peri, would gather a dozen teenagers together and pass out copies of a poem. Either she or a student would read it aloud, and then we'd talk about it. She'd ask what people thought, what lines they particularly liked or didn't like, what they got out of it. Frequently she'd ask us about a specific line, or point us at an unusual metaphor, and if the poem had a specific set form she'd mention it. It was equally valid to say we didn't care for the poem or to say it completely changed the way we looked at the world. Then we'd have the last half of the class to write. We could write something inspired by the poem or something completely different.
A substitute teacher came in one day with a completely different style, although she was similar on the surface. She brought in The Fish by Elizabeth Bishop and had us read and discuss it. She asked us what we thought a particular line meant the way Peri would, but when someone answered she'd say "Yeeees..." in that way some teachers do which means "That's true, but it's not the answer I was looking for. Siddown and shuttup." I still dislike that poem. (Although that could be because it's just not a good poem. It seems to be required reading in various poetry classes, though, so either I'm missing something or it's just a good poem to ask students to interpret a specific way.)
Being asked my opinion of a piece of art and then being told that was incorrect was not a pleasant experience. If the entire class had been led that way, I'm sure I would not like poetry today.
If you plan to be a poet, it is certainly good to be familiar with different literary devices and forms. But I firmly believe that the best way to do this is not to have a textbook lay out metaphors and explain how they can emotionally manipulate your readers. You'd do better with a good dictionary, a thesaurus, possibly a rhyming dictionary, and a good listing of different poetic forms. Bad textbooks give you perfunctory definitions, such as "a haiku has three lines of five syllables, seven syllables, and five syllables." You'd be much more likely to write good haiku if you spent a couple months reading Basho and figured out for yourself what's important in a good haiku. After all, that's how poetic forms got decided on in the first place. Someone read a lot of poetry and decided that if it had this, this, and this it was a sonnet. If it had this, that, and the other it's a pantoum. Definitions of forms are useful, but not as a be-all and end-all.
A poem can have the most beautiful form you like, and still be a bad poem if the content, the emotion, the (dare I say) universal draw isn't there.