Every Linux distribution out there (and, indeed, every OS in general) has its own set of unique quirks. One that I've just come across in Antergos with Gnome 3 (I have no idea if it's an issue with other desktop environments) is that the power saving features weren't turning off the screens like I had specified in the settings. The screen was also not locking as it should.
It took a bit of searching around, but it turns out that I'm not the first one to have this issue.
Installing Light Locker Doesn't Work
Before I state how I fixed this, let me first say that light locker simply doesn't work. I've seen this suggested as a fix for this problem a number of times, but every time you close the light locker settings it loses all your preferences. This is a known bug and renders the program pretty much useless.
What Does Fix It
So, the base problem seems to be with the default display manager in Antergos, which is
lightdm. This means that the problem can be corrected using an alternative display manager.
Since I'm running Gnome, I opted to just use
gdm as my display manager, but presumably any display manager, other than
lightdm would work. This can be done by simply running a few commands:
$ sudo pacman -S gdm $ sudo systemctl disable lightdm $ sudo systemctl enable gdm
After a system reboot everything should be working again. NOTE: I personally had to reinstall the
gnome-screensaver package to get things to work, but I think it ended up getting removed when I was trying some other things to resolve the issue, so you may not have to do this.