Someone who knows about drop spindles! Yay!
I can't imagine a motorized drop spindle, but it would be very very cool.
I'd think that a hollow bell shape or similar, with a battery, a small motor and a flywheel would do. Just hold it to stop rotation.
It's sort of funny... You wouldn't have been surprised, had I been female I gatherr (?) - Ought I be surprised that you, as a woman, play around with motors?

To really throw you off your rockers, I could tell you that my present (third) sewing machine is utter crap, but I wouldn't like to be without it anyway

Sorry, but I'm just a little confused about how you are obviously rooted deeply in rock hard gender roles AND breaking them at the same time

Wasn't them womenfolk that invented the sewing machine and similar machinery in the first place either

Yet another paradox is, that you ask for assistance, but are reluctant to give the needed info?
If you're afraid of your competition getting wind of your super spinner, you're welcome to drop the numbers and such in a PM, if you feel comfortable with that, but engineering works on numbers, not adjectives, so I won't be much help without this info - it'd be like if I asked you to knit me a sweater tight enough to show my abs but not too tight either and then
not reveal my measures.
Those Hansen spinners are ridiculous expensive (and look like something that a 14yrs old kid made in wood shop class) - shouldn't be hard to make it under a third the price and 3 times as purdy.
Brushless PC fans are very small and weak motors, but the same techniques are used in fans that could take off an arm and about noise... it's not the motor in itself that makes the noise, it's bad balance, bad bearings and in the case of a fan, wind noise when the blades disrupts the air flow. Noise in a gearbox is likewise a matter of too sloppy engineering, bad materials, too wide tolerances and such - professional minigrinders (the pro version of a Dremel) doesn't vibrate in your hand when going 40kRPM (or faster) and it is barely audible until you start grinding at something.
I have gearhead motors that are so silent and well balanced, that you have to touch the spindle to know if they're turning.
With a good motor, I'd think a reasonably well build spinner could be made to max out at around 30dBA (perhaps less), if your mechanical provess is up to it. Mind you, I have no hard and fast idea of the power it takes to spin or what speed/torque you need, as I have never had any hands on, but I'd think that the Hansencraft never really used all of its 17W - I'm just guessing here though, based on spinning stuff like string and metal wire.
A stepper motor with enough power would be fairly silent, although quite bulky, but why not do away with old thinking and use direct drive, rather than a "belt" drive?
You might wanna look into pancake motors, they're quite powerfull for their size, but need a driver like any other BLDC motor.
I don't know if the Hansencraft motor is indeed coreless. I see no need for using a coreless motor (quite the contrary actually), as the main feature of coreless types is that they respond very quickly to speed changes, as their core is "just" some copper wire on a light plastic drum, so the (moving) core don't have the inertia of a conventional iron core motor. Spinning probably needs rather slow/smooth speed changes I'd guess.
The control electronics of the Hanencraft is very simple as well - anyone with a Robotics 101 under their belt should be able to make a similar control in their half sleep.
[...] I'm building this spinner for a spinning competition,
I imagine the spectators jumping up and making the wave, chanting encouraging and similar ;P
How much are the tickets... Hehe

Sewing machine motors?
1) They are AC motors, and I'm too young to die 
Ain't we all?
Happens anyway, my best friend, from days past, bought the ticket a month after his 18 year birthday.
But to die from a properly encapsulated AC motor, takes a Darwin Award nominee so...
2) too noisy
Most of the noise in a modern sewing machine, comes from the mechanical actions happening downstream from the motor.
3) too big.
Sorry, didn't know you were running out of space, since you didn't post any numbers

I know there are men out there making these sewing machine motor-based spinners for their wives.
Uh oh, back in the predefined gender roles *G*
All I can say is that they really must wish their wives dead or something!
The number of times I have read forum posts by women saying "My husband made me a spinner from a sewing machine motor. It's getting hot, and so is the pedal. What could be wrong?"
OMG!!!
Come now... Nobody ever died from a
little heat (Indian wives and kitchen furnace "explosions" is another thing though).
I think that a more realistic gender differentiator is noise... Men
like machinery to be loud - the guy with the loudest toy wins - while women are like... "You don't need those large speakers and could you please turn that ghastly music down to -6dBA immediately" - I bet that's why there's so many broken marriages

There is no reason in this day and age of micro DC motors that we girls should be subjected to sewing machine motors for spinners!
One reason I can think of is, that you girls aren't telling us primitive Neanderthals what you need, so we just have to guess and our Martian reptile brain will always prefer a just over-kill over just a kill, just to be on the safe side - women everywhere have heard it numerous times, I'm sure, but they still think men are mindreaders (primitive, but with a superior ESP

) - If women actually told their men how they wanted it, in measurable units, I'm sure the men would step up to it

[...] I got another small brushed motor (not geared) which comfortably does just over 2000rpm, but it's not as fast as I would like.
If you use a belt drive, it will have a gearing worth the difference of the driving and driven "wheels" diameters.
Say the pulley on the motor axle is 20mm and the one on the bobbin is 100mm, you'll have a 5:1 gear reduction and the 2kRPM is down to 400RPM (which I'd think would be pretty fast when you have to feed the fleece in a steady flow, but what do I know).
Telling
what you'd like, rather than what you
don't might spur a more helpful response and the same with pulley diameters and such.
I'm won't mention specific torque figures either because it took me a lot of trial and error to figure that stuff out for myself, and again, it's for a competition where practically everybody will be using the spinner linked to in the video above
. Mine is different and has a couple of features that commercial spinners don't have but which are really important to me.
As I said, you could PM me with the figures and I promise you that I won't leak a word.
[...] for an RC motor you need an ESC and that would have to be housed, which would make the spinner build bigger
ESC stands for Electronic Speed Control, just like there is an ESC in the Hansencraft.
I'm also not too keen on the idea of LiPo batteries which could unexpectedly explode inside the my spinner
). I'm really just looking for a little 12V DC motor that I can connect with an adapter in my living room.
You don't
need to use LiPos with a BLDC, any battery or a mains adapter is fine, as long as the voltage and current is matched - and the dangers of LiPos are greatly exaggerated - just think of the billions of cell phones in use world wide - how often do you hear of someone getting their head blown of by their phone?

You can always use LiFe cells if you decide to make a portable spinner, they're considerd the safest of Lithium cell chemistries

In terms of motor specs (RPM, torque), the brushless motor fits the bill. And so does my little brushed motor (even if it's a little slow for my liking).
So, your problem is solved with the brushless? Great, happy construction then and please post some photos of the finished version (when you've won the gold and the competition is left behind)

I'd still like a link if you could be persuaded, as I both use a lot of motors myself and does a fair bit of consultant work on motors as well, so I'm allways interested in new stuff.