The reason for the din of not drinking wine during the 9 days is given as a way of remembering the lack of nesachim due to the lack of korbonos, one of the results of the churban (not due to aveilus,which has no issur yayin).
Although it is technically permissible to drink mitzvah-wine, the minhag is to give the havdalah wine to a child to drink—the age must be old enough for chinuch in berachos (you are being motzie the chld in the borei pri hagofen),but not old enough to understand about being misabeil on the Beis HaMikdash(this is usually considered to be between 5 and 10).
If there is no boy of that age, then the person making havdalah should drink the wine himself(rather than giving it to a girl or a child below or above the age group given).
It is brought down that HaGaon R’ SZ Aurbach zt”l drank the havdalah wine himself, rather than taking a chance of berachah levatalah (if the child is too young)or of there being no point(if the child is too old).
Surely we mourn the churban Beis HaMikdash.Surely we would recoil at the thought of aiding in it!
And so let us always recall and be cognizant of the warnings of so many Acharonim that lack of respect in a Shul, idle talk, and talking during davening was the cause of the destruction or desolation of all too many Botei Knesses.