What's the recommended tool for turning a toothpick? (please don't say skew....please don't say skew...) << fingers crossed.
I would use a SMALL skew.

(I often use a 1/4" skew for small things or small parts of larger things.) Or, since you live in an unknown location I don't know if it's too far, but if not, come visit for a day, become proficient on the skew, and we can make some toothpicks and you can go home and make some more.
But if you just want to show the toothpick for scale, it would be easier to forget making one and just turn the tiny things on the end of a small, short blank held in a chuck, and show them along with a real toothpick, or with a penny like I do, or something else, for example,say, inside or around a woman's diamond ring. Or next to a pencil point, etc.
Turning from a small, short blank, I usually use a 1/4" detail spindle gouge, a narrow parting tool, some tools made just for small things, and a couple of small negative rake scrapers. That and some 600 grit sandpaper will do almost everything. I have chuck jaws that will hold down to a 3/8" round blanks.
Note that if want you show pieces along with a toothpick, maybe they don't all have to be made from the same wood - you could easily show things from other woods and have the toothpick can be there just for scale.
If you don't have small pin jaws for a scroll chuck like I use with Novas, you can take the jaws off and grip a small round or square blank with the jaw slides themselves.
Also, if the goal is to learn to make really tiny things, the diameter to more than the toothpick, I highly recommend first turning some larger small things for experience/practice then work your way down.
Note that there are other tiny things you can turn as well as bowls, goblets, etc. For example, I like to make tiny "magic" wands! This one from Ebony is about as long as the diameter of a penny. (not to scale with the goblet!)
JKJ