Instead of spending upwards of $30 on a new stand and placing yet one more thing (that’s only 0.001% broken) in a landfill, I did some research.
Using a pitch gauge, I found that an M5—0.8 × 20 mm screw will keep the neck of the stand in place. Home Depot carries the screw, whose SKU is 887480028287, for $1.05. Here’s a link.
The trade-off for using this screw is the forfeited ability to use the stand’s one-handed adjustable feature. (To adjust the height, I will need to unscrew, re-position, and re-screw the stand’s neck.) No matter, I never used the adjustable height feature.