The docs, don't specify the need to install the pwa to be able to use the notification feature. However what I suspect is happening in your case is that Chrome is not giving priority to notify to the notification that you are sending without installing. What I mean is that you might receive your notification on the regular wakeup cycle of Chrome, and not as a background task. (But this is just a speculation)
Another common scenario that happens a lot, trust me :-), is that you forgot to give permission to send notifications in the first place.
Regarding remote debugging, refer the docs, to get it setup on Android. As a lot of the online tutorials are a bit out of date.
Note: I found an article online that shows a notification received without installing on Android, here is its link, it might not be very helpful for your case but check it out you might figure something out.
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