Newborns do not temperature regulate well, and need to be kept warm for the first few weeks/months. Equally though, not too warm, since they can't cool themselves down. The elderly also to do not thermoregulate as well, and tolerate extremes of temperature poorly. I guess people with v low body weight, chronic illness etc, might also be affected by temperature extremes. The rest of us do fine in a relatively wide range of temps: its called homeostasis.
Personally I think 60 at night is completely fine. In Australia we have poorly insulated houses which don't hold the heat. Most people don't heat their houses at night as it doesn't really get very cold (well not like northern US states at least). Houses commonly get down to 45-50 degrees overnight and I don't think we think anything of it.
For example I live in a frost free area, coldest it gets is 40F overnight. Before I fixed my house insulation up in the winter it was 45 degrees in the kitchen in the morning eating breakfast. Unpleasant yes, did I worry my kids would die of pneumonia, no. Now in winter the house will drop down to say 50-55 overnight which we find fine. I wouldn't dream of leaving the heat on to keep it at 60 all night. My kids have lived very healthily in this sort of temp range (>50 in the house) all their lives even when they were newborns.
If you were going to change anything, I'd do it gradually over months or even a year or two: there is a period of adaptation. One of my friends is an early childhood teacher who said she had young preschool kids just arrived from Vietnam (much hotter and more humid) shivering in their parkas all summer for quite a while until they got used to our temps.
edit: clarified an ambiguity