I'm not in the US, but as an immigrant myself I can say it's incredibly stressful to be on a working visa during job changes and during a period of unstable economy. Especially if you have family.
To be rational, I work in academia and I shouldn't need to worry too much about job safety and/or visa applications, but I still semi-regularly woke up at night from nightmares of my visa expiring, not being renewed, suddenly being deported, or similar stuff.
I got my permanent residence earlier this year and all of it stopped. It gives you some sense of stability/security. It also makes one feel a bit more accepted in society by not needing to leave it if a work agreement (for whatever reason) ends.