The Aish link you brought is very good in setting the tone for what you know and want to find out, so thank you.
In regards to borei moreih ha'esh, the Gemara brings the idea of waiting to say the bracha contrary to saying it right away just as a minhag, not too much reasoning there (see Pesachim 54a, R' Yehuda holds and R' Yochanan says halacha is like him-R' Yochanan usually the authoritative Tanna as far as psak from Gemara goes and we see that follow through for Rambam Hil Shabbas 29:24 and Tur Shulchan Aruch 296:1)
In Shulchan Aruch, the Rema notes the minhag of pouring the wine first as a sign of a bracha (for the week) and specifically over the meorei haesh-- no reasoning.) The first reasoning that I found was is in the same siman of the Shulchan Aruch HaRav (Seif 5) quoting the Levush (271:10) that pouring the wine to put out the candle signifies the candle's use specifically for the mitzvah of havdalah and brings in the name of the Rema (mentioned earlier) that we rub our eyes with the wine to signify our enjoyment of and love for the mitzvah.
Seemingly this wouldn't be an opportunity without the candle for the wine but my Rosh yeshiva said this wasn't the reason, just the way the minhag is fulfilled through the meorei ha'esh and bsamim being a part of Havdalah. He further mentioned that it's logical (and potentially why it's hard to find mefarshim say so explicitly) that in general, saying something over wine in Yiddishkite signifies an importance ceremonial emphasis of kedusha as we see it used for kiddush/Havdalah for Shabbas and festivals, kidushin of the chupah, Birkas hamazon with a zimun. It in a sense anchors down the mitzvah in a mehudar way so how much more so when you anchor the two other related brachos related to Motzei Shabbas.
Fun practical minhag some have as an outcome of saying all three brachos together is the ability to light the candles required for melaveh malka (as in Kittzur Shulchan Aruch 96:12) with the Havdalah torch before putting it out with wine. Hopes this helps.