This quote is mainly arguing against the attitude that says "you have feet therefore your shoe problem is a non-problem, don't even bother feeling bad or working on it".

That may be, but the actual context of the quote it's arguing with is quite different, on a couple of fronts.

Harold Abbott, the author of the original 1934 couplet ("I had the blues because I had no shoes / Until upon the street, I met a man who had no feet"), wrote it to memorialize an encounter with a happy legless man, at a time when Abbott was dead broke and depressed. (Abbott was not actually lacking in shoes, nor the man only lacking in feet, but apparently in those days people took their couplet-writing seriously. ;-) )

Thing is, at the time he encountered the legless man (who smiled and said good morning), Abbott was actually walking to the bank in order to borrow money to go to Kansas City to look for a job. And not only did he not stop walking to the bank after the encounter, he decided to ask for twice as much money as he had originally intended to borrow. He had in fact raised his sights, rather than lowering them.

That is, the full story is not anything like, "other people have worse problems so STFU", but rather that your attitude is a choice, and there are probably people who have much worse circumstances than you, who nonetheless have a better attitude. Abbott wrote the couplet to put on his bathroom mirror, as an ongoing reminder to have a positive outlook and persist in the face of adversity.

Which is quite a different message than what Noah Brand's snarky quip would imply.