I don't know why Barack Obama [almost] never wears a flag pin, though I can tell you why I am uncomfortable with wearing one.
If you check the US Flag code,  http://www.usflag.org/uscode36.html , you will see that using the flag as an article of clothing is considered disrespectful save in two circumstances--well three, but I propose that it is really only two:
1) as a service patch for military and for civilian workers such as firemen and policemen;2) a carefully designed and authorized "flag lapel button" only to be worn by immediate family members of the armed forces during a time of war or hostilities;3) the flag pin, which can be worn by anyone, as long as it is on the left lapel near the heart.
I've always felt uncomfortable about the last,  because it seemed so odd to forbid clothing except in special circumstances, while giving a flag pin a pass, and I just re-read the flag code. The flag lapel button is mentioned eight times in the flag code, always with the article "the." Careful guidelines are given as to who is authorized to wear it, who is allowed to make it and what fines can be levied against its un-authorized manufacture ($10,000 in 1942!!!), but no explanation is given about where to wear it, save "lapel."
The flag pin, on the other hand, has no guidelines whatsoever save where to wear it. There are two possibilities:
1: the flag pin is meant to be made by anyone, designed by anyone, worn by anyone, for any reason, even though it isn't mentioned anywhere else in the flag code--you can even design it to look like the special privilege flag button without risk of fine if you like;
or
2: "pin" is a typo and the code should read "the lapel flag button" and only authorized people can wear it or make it as it says elsewhere in the code.
My intuition is that my original uneasy feeling about wearing flag pins is correct: they are inherently disrespectful, according to the flag code and it is only a typo that allows them to exist and be worn without remark.