Scaling or Thinning Bones I have created a dragon mesh. Eventually, it will fly.

Right now, I am trying to add a bone to the mesh (near the "elbow" of the wing), but it is a huge piece of mesh and the armature bone becomes huge, too i.e. it doesn't fit inside the wing. Please see what I am trying to do below and how it currently looks. Note that I am new to Blender.

Maybe it isn't a problem if the bone sticks out; I don't know. In the end, when animating, can I hide the bones and only see the mesh? If so, there would be no problem for the bone size, I guess. Tried to look this up on the internet, but could not find exactly this issue addressed. Thank you in advance.
An Armature never shows up during a render. If it sticking out is annoying while modeling, you can choose between five options for its shape. To change, select the armature and go to the Properties panel > Skeleton tab > Display dropdown and choose. I would describe the five, but it would take a while, so I'll just say that I believe Stick is the smallest.



Note that the image is a cropped version of one belonging to chichige bobo in this YouTube video.

Lastly, just thought I'd add that Octahedral bones (what is currently displaying) do stretch out like in your third image, but only when using a Stretch To Constraint. You would not want to employ this for just basic rigging.