Thanks to @niceSprite for the pointer. I was able to recreate a new macOS account having transparency, so I started going through my list of customizations on my main account.
It was a defaults issue with the key "NSRequiresAquaSystemAppearance".
When I upgraded to Mojave, I lost the option to have a dark menu bar with light windows. I tried dark mode and didn't love it, and followed this article to get back my pre-Mojave menu bar color: https://www.tekrevue.com/tip/only-dark-menu-bar-dock-mojave/
So I did
defaults write -g NSRequiresAquaSystemAppearance -bool Yes
It wasn't perfect, so I reverted back with
defaults write -g NSRequiresAquaSystemAppearance -bool No
That caused the issue. I re-read the article and found I should have deleted the key entirely. So I gave this a go:
defaults delete -g NSRequiresAquaSystemAppearance
After logging out and logging back in, transparency was back in Alfred. I don't know what this actually did that messed with Alfred's transparency specifically, but this was a resolution for me. Hopefully it will work for @niceSprite as well and we can call this resolved.