Because it's often a lie (at least in my experience). When I worked in defense contracting people loved trotting out stuff like "the work is stable", "you can't get fired from a government job", etc. all while dealing with furloughs, government shutdowns, geographic consolidation of work to different states, contracts changing hands, companies being acquired, etc. Employment in the US is at-will and nobody is going to guarantee you a job, especially these days, and employers at the lower echelons of the industry are very fond of selling the dream of "stability" by sowing fear of the unknown to prevent attrition while building a culture of malaise and low compensation.

I'm not saying that one should work exclusively for seed-stage crypto startups that sell blockchain-verified probiotic dog food or anything, but at some point you should critically examine whether it makes sense to be someplace that's 10 years and tens of thousands of dollars behind the curve just because it's "stable" when it so obviously isn't.

You’re likely better off getting a job that pays better, but being laid off more frequently.

If the layoffs never happen, you’re in a massively better position. If they do, the extra income still puts you ahead of staying at a stable, but poorly paying job.