I wrote three paragraphs, but I guess since I didn’t directly address the magical thinking of the effect of from reflector brackets on self-steering, none of it counts. I’m also not going to go out to the garage and tip my bikes over to see which side the handle bars flop to.

Here’s a data point for you:
I have two similar bikes: a Cannondale F-1000 and a Klein Pinnacle, both aluminum frame, 26” wheel MTBs with ~700mm wide bars.
The Cannondale has a 140mm stem and is nearly impossible to ride no-hands; only if you’re going fast, straight, and you hold really still.
The Klein, OTOH, will go for blocks; you can pedal, get a drink, take pictures, go around curves; I’ve never had a bike that stable.
It has a 50mm BMX stem.

It’s not the stem length.