1. It is infinite & inexhaustible. Well, not really, but close enough. You couldn't in a lifetime read and understand enough poetry to exhaust what poetry is, or even a few traditions of poetry. You will never run out of things to do or to study. Aside from a few canonical authors, poetry is not well studied. Even the study of canonical authors is problematic and often superficial.
2. It is demanding. You have to step up to an intellectual & poetic level that is adequate for the material being studied. You will get smarter by doing this. If you are smart, you will be able to exercise your complete intelligence with this genre, including your musical and visual abilities, such as they are.
3. You can't practice content mining. You can't deal with poetry by "content mining" or hermeneutical strip mining, simply reading it for its overt treatment of race, class, gender, etc... You can still deal with these and whatever other issues you want, of course, but your approach will have to be sideways rather than direct.
4. Not everyone does it. Spires once asked me in an interview why I didn't study the novel. Well, everyone else does (or used to at that time, now it would be film.) Why should I do what everyone else is already doing more or less well?
The disadvantages are that you probably won't be able to do it very well. You might not rise to the challenge. In spite of poetry's inexhaustibility you will produce a reading of a canonical author that is not that different from that of other critics. You will content-mine in spite of everything.