My iPhone won't show any Discord iOS push notifications, yet it will show them from other apps
This is a Discord “feature” that you can’t turn off. Discord will not send mobile notifications if the desktop app is open (official staff response). Here are some community posts requesting Discord to change this feature:

Allow 0 minute setting for &quot;Push Notification AFK timeout&quot;
Displaying Discord Notifcations on the Phone and the Computer at the same time.
push notifications not sent on android

Discord tries to avoid being annoying in the hopes that it reduces the chance that you uninstall it by not sending push notifications when it thinks you are active at your computer. There is a setting to set the number of minutes Discord waits after you become inactive at your computer prior to resuming push notification in the desktop app under Settings / Notifications. You can only set this down to 1 minute. So you won’t get push notifications until Discord thinks you’ve not been at your computer for at least a minute. But apparently Discord likes to think that many people are using their computers even when they aren’t much of the time.
People report that fully quitting the desktop app of Discord sometimes triggers Discord to start sending push notifications to your phone immediately. You can still use Discord from a web browser without affecting push notifications if you don’t need any features from the desktop app.