I'm moving my note to a full answer because I think all the answers are incorrect. They stem from assuming the writers were completely perfect omnipotent beings incapable of creating a plot hole, and then coming up with "what if" and "maybe" scenarios to fill in the details.
The actual answer is simple:
The writers did it so they could put his kid on the ship (and add tension by showing the suffering of people). It doesn't have to "make sense", because it doesn't. Just because Earth falls, doesn't mean humanity falls--this was directly addressed by Guinan:
[TNG: The Best of Both Worlds]
Guinan: This isn't the end.
Capt. Picard: You say that with remarkable assuredness.
Guinan: With experience. When the Borg destroyed my world, my people
were scattered throughout the universe. We survived - as will humanity
survive. As long as there's a handful of you to keep the spirit alive,
you will prevail - even if it takes a millennium.
Bringing the civilians--as opposed to launching early escape pods / shuttle crafts / teleporting them to a station, planet, allies, and traders that frequent Federation space! ... Bringing them into a suicide fight makes no sense--unless you want a reason for Sisko to watch his wife to die at the hands of Borg Picard--setting up a series long arc for Sisko to both hate, and eventually learn to let go and forgive, Picard.
Bringing civilians into a suicidal battle with a ship that has killed many ships before you got there, and continued to kill ships, without once thinking "My gosh, what about our civilians?" is so incredibly out-of-character for the 1990's-era Federation that it can only be reduced to one thing: It's a plot hole. In PLENTY of the TNG episodes they explicitly deal with this exact problem and jettison the saucer section, or put medbay on standby and move civilians to the most protected sections, and so on.
"Overconfidence"? That's silly. You're fighting possibly the largest ship ever seen (which had destroyed multiple outposts already), and assembling the entire star fleet (which as far as we know has never been done before) but we're still confident enough to not worry about civilians? Scrambling all warships (and improvised warships) and "overconfidence" are not two compatible ideas.
"Humanity is doomed if we lose, so let the kids die"? Directly addressed by Guinan. Humans are survivors. They wouldn't just give up and die if their fleet dies. That goes against the core of all Star Trek canon--humanism, that humanity (and not needing "gods") can achieve and overcome anything, including conquering the stars. And Andromeda (while not "technically" star trek) basically is set on the premise of the Federation collapsing--the ultimate failure--and yet with a single guy surviving, he still goes right at it, to attempt to reunite the commonwealth.