I got the work done with the neck.
Nicely done! That is a fine outcome for your first cracked neck repair.
Poking tools, guitar stings, etc., into the crack will help distribute the glue, but it is also risky because you may displace splinters or narrow wood fibers, which can prevent the crack from completely closing.
When I repair a 'closed' break like this--one that's long and deep into the wood but one I don't want to open up--I use a little trick I came up with many years ago to get glue into the joint: compressed air.
I clamp the neck to the bench and use another clamp to gently hold the break open without causing further damage. Next, I lay a generous bead of glue (un-thinned Titebond II) along the crack, and hold the nozzle of a hand-held compressed air valve in line with the break with one hand. Before giving it a few quick short toots I cover the work with a small shop towel held in place with the other hand.
The shop towel is important because without it, the glue will splatter and get on stuff in your shop that does not need gluing. It is also important to wear eye pro because squinting or shutting your eyelids is not safe enough if glue comes flying out from under the towel.
So give it a toot or two, lift the towel and reposition the nozzle, cover again, and one or two more blasts. Lather, rinse, repeat. One thing to do is use a fingertip to push runaway glue back over the crack and continue. Like most any glue job, you'll waste more glue than you use in the joint, but this is good. You have about three minutes of work time, which is plenty of time for a break like this.
Gently close the break with the padded clamp(s), and have some slightly damp towels on hand to wipe up the glue squeeze-out. A few Q-Tips are also handy here as well.
All through the inspection, handling, and gluing procedure, it is critical that you be very gentle with the break, especially the very fine edges where the
finish is broken. Ideally, you won't make contact with this fine edge and knock off pieces of finish that you have to fill later, or worse--push pieces of broken finish into the crack. Anything that does not belong in the joint and any wood fibers that are misaligned with the sound wood in the joint will prevent a nice tight closure.
If you don't have an air compressor, canned air ("computer duster," etc.), will work too. Use the little red straw in the nozzle.
You can crack a short piece of wood or a section of large dowel like a broom handle and practice this procedure to get a feel for it.