YSK that most of these have an expiration date (and should have an expiration date). Checking the box says trust this device that I'm currently using, for a while, thus:

• if you use another device to log in (e.g. switch from mobile to desktop browser), you will need to do it again, because security is tied to your ownership of the device, and
• if you continue to use the same device for more than--usually--a month, you'll be asked again because there's a possibility your device got stolen or compromised in the meantime.

In short, most of the time you'll have to do this once a month, or more if you're using more than one device, and that's nominally to protect you.

However, yes, a significant portion of these are just implemented by dummies, and don't work.

The particularly annoying ones for me are the ones on bill pay sites. If they work for a month, that doesn't help me. I only pay my bills once a month.