Sure. Great idea. Brilliant. Why wouldn't we welcome it?
The comparison to "getting colder" isn't really valid, as it is not the only option.
Most people would like things to stay the same. Perhaps we should be looking at what temperature would be the most beneficial to the planet as a whole.
If we were to create a new earth, what would be the ideal temperature, how would we manage it, where is that in comparison to where we are now, how can we get there.
Is it a dangerous question? probably.
Earth itself will do just fine at any temperature that's below literally evaporating. Various life on this planet has evolved to thrive in wildly disparate climates, and short of complete eradication of an ecosystem, surviving organisms both tend towards better suitability for their given environment and adapt their environment towards better suitability for them (both with limits). We humans should be looking at what temperature ranges would be the most beneficial to human society as a whole. Sure, that's incredibly self biased, but as a human, I contend that's an incredibly useful bias to have.
Attempting to adapt the environment will have unforeseen, probably serious, consequences. I for one hope we do not try, or have no choice but to try, geoengineering attempting to avoid catastrophe.
To your second point I've always preferred climate change as a term over global warming as some areas will end up cooler, others more changeable. Especially if one of the major ocean current systems slows or stops.