287
Answer: The sources leave room for much hypothesis but little conclusive evidence.
The Shulchan Aruch (Orach Chayim271:14) deals with the way(s) to fulfill the requirement that a m’lo lugmav (enough to fill cheeks – approximately 2 fl.oz) of the Kiddush wine is drunk. Some say that one person has to drink the whole amount; others say we can add up that which different people drink. The Shulchan Aruch points out that either way, the choice way to perform the mitzva is for everyone to drink. It is sufficient for each person to have a small amount (Taz ad loc. 17), and if their drinking interferes with one person having a m’lo lugmav or leaving wine for the next day, the idea of everyone drinking is waived (Magen Avraham 30). Yet it is important enough to delay the mekadeshbetween his beracha and drinking (see Shulchan Aruch, ibid. 16).
The Shulchan Aruch’s source (see Beit Yosef) is the Rosh (Pesachim 10:16), who explains the goings on in the gemara’s about people drinking Kiddush wine: "Although they are not required to drink, still it is a preferable mitzva to drink." He does not offer a source, or an explanation, nor does he mention if it is a special mitzva regardingKiddush, which is the gemara’s context.
The Rambam (Shabbat 29:7) says that after drinking a m’lo lugmav, one "gives to all the members of the group to drink." The Mirkevet Hamishneh (ad loc.) looks for a Talmudic source for the Rambam (who rarely includes a halacha that lacks one). He points to the gemara in Berachot (51a) that lists things one is supposed to do to enhance a kos shel beracha (cup of wine used in a mitzva context). Rav Avahu mentions ten things and then that some say to send it to the members of one’s household. R. Yochanan argues that only four of the practices need to be kept. The Mirekevet Hamishneh says that R. Yochanan reduced the ten to six but did not take issue on sending to one’s household. If this is the source, then it should apply to all cups ofberacha. Indeed, the Shulchan Aruch (OC 190:40) says so regarding wine for Birkat Hamazon, and it should ostensibly apply to Havdala. The logic is that drinking the wine bestows importance to this mitzvacup (Darchei Moshe, OC 182:1).

Bemare Habazak - Rabbis Questions (596)
Rabbi Daniel Mann
197 - Listening to Laining During Shemoneh Esrei
198 - Who Drinks Kiddush/Havdala Wine and Why?
199 - How Can We Say Things of Minhag Before Kiddush?
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One can suggest positive reasons for drinking specifically atKiddush, which will also explain the minhag. Some claim that the obligation to make Kiddush over wine has a stronger basis than other cups of beracha (see Encyclopedia Talmudit, v. 27, col. 510). Also,Kiddush is connected to the meal in which all are partaking (there are different explanations of the connection). Since it is positive to drink wine during the meal (Shulchan Aruch, OC 250:2) and when one drinks wine at Kiddush, he is exempt from a beracha during the meal (Shulchan Aruch, OC 174:4), it makes sense to start drinking atKiddush.
In any case, while halacha does not obligate everyone to drinkKiddush wine nor forbid it at Havdala, your observation has both sources and a variety of possible explanations.

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