Depends on how you define "Cold-Hearted"

If you define Cold-Hearted as no longer caring, no.

If you define "Cold-Hearted" as skeptical about (promised?) happily-every-after relationships where your positive support for others is returned in kind, Yes.

I disagree. Perhaps they're giving that "caring" to people who don't deserve it or who aren't going to take care of it. If they love someone, and that someone treats them like garbage, they can still love the someone but get away from them. If they stay, then that's what burns them out. There are people in life who are "vampires..." they suck the very life out us. If we stick around with those people, we are drained dry. There are others who build us up and call out what's good in us, and encourage us to be better. Those people charge our batteries. To the decree we can, we also need to make good decisions about the people we get close to.

Not all cold people got there that way though. Some are just born jerks.

Nope, I grew up with a whining mother who used tears to manipulate.

If someone needs help, I'll help.

If they just want to cry over the same thing over and over again with no plan to take action, just don't bother me with it.

People are overall more trouble than they're worth.

Very true indeed