Modern server hardware can hit very low power states at idle just like modern desktops can. Most servers are also headless, so you’re not wasting power on a GPU. This means that you’re looking at a lower power bill if you’re not doing much. Likewise, you usually pay most for egress, and at low traffic that’s not much of an issue. Physical hardware can scale down exceedingly well. The main thing I would argue is that one own the hardware rather than rent.
It's hard to scale down the cost to the range of $4/month. I've seen offers for rented dedicated severs in that price range, and it's Core2 and similar.
Anything you own in a colo is going to be more per month too. When I had connections where I could pay for a static IP, that was usually $5/month.
I'm now renting a pretty low end server, but it's $30/month. Way more everything that I need, but it's nice. And they didn't drop support for my OS while increasing prices to improve support or something. (Although I did have some initially flakey hardware that needed to get swapped)
When colocating, contracts tend to pay for power at a fixed price based on provisioning (which has to be more than the max power that will be used). Power savings are generally irrelevant for the customer paying for colocation despite being a good thing for the data center.
