I know someone who has done it – he uses his old Explosif for commuting duties. He does, however, have a set of STIs on it as he runs it geared.
There’s no real reason why this could’t be done, though as people have pointed out, the TT length on an MTB means that you’ll need a short stem.
You will also probably find that, like most people who ride drop bars, you’re not actually in the drops that much, instead riding with your hands on the hoods or on the tops. However, the option to change hand position is excellent, and a great reason to use them.
For a bit of compromise, look at wider drops (Rivendell, One-one Mary) and also cross drops, which sometime flare in the drop. On the road, I favour a shallow drop bar – you get the efficiency of drops without having to remove a rib like the old Belgian deep drops require. Have a look at the FSA Omega compact when available again.
Otherwise, you need to think about gearing. Off road will need something low-ish (though I note you’re planning on using the bike for family rides). On road, something bigger will be necessary – say 68-72″, depending on how hilly your commute is.
Also, think about wheels – if you’re only ever running reasonably skinny tyres, then maybe a 700c wheel would be a better choice. You could have the rear built up on a MTB hub. If there’s enough clearance in the frame (you may need a 29er fork), then you could probably swap between CX and regular road (say 25c) tyres.
Just a few thoughts….