Oh yeah, I should update this as I've discovered the issue/answer. To answer your question, though, the apps are compatible, I believe. I tried it with some pretty mainstream third party apps such as Slack and Skype (in addition to lesser known ones).
But here's what I've determined. Nothing was broken, it's just the native behavior of macOS. I find it to be a little strange, but ultimately in most cases the whole sequence probably happens before anybody notices and thus it's not an issue.
Basically, when you download a third party app that has notification capability, it still will not appear in your notification settings until the app has actually attempted to send a notification. Once that's happened, then that triggers something in the OS and the settings are available in your System Preferences > Notifications.
For example, while I had Slack on all day, my Slack settings called for notifications only when someone direct messaged me or tagged me in a channel message. While I had channel messages throughout the day, none of them met those criteria, and so Slack had never attempted to send me a notification. I was proactively trying to set them up to my liking in my preferences and became confused about why I couldn't, but ultimately I hadn't actually missed any notifications.
Once someone sent me a direct message, I did receive a notification banner, and immediately thereafter, I checked my System Preferences > Notifications, and Slack was then available in the menu for me to modify.
So much adieu about nothing really. Again, I do find the order of behavior odd, but now that I understand how it works, future frustration will be avoided.
