Is it possible to have an SMS app that can block some kind of SMS (i.e. where text contains "XYZ" phrases), but I don't want it to be set as the default SMS app?
I've seen something like SMS Radar but don't know if that can do the task.
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This isn't possible as of Android KitKat (4.4). Only the default SMS app is able to stop the broadcast of the SMS message to the other apps.
That said, since Android N, you should be able to use blocking to block SMS and Calls from a particular telephone number using Android N Call Blocking.
Note that some Carriers/OEMs may actually disable this blocking feature or make it harder to get to.
There is a Content Provider and API starting in Android N that would allow apps to programmatically block calls and texts, you need to either be the default SMS or phone app, a system app, or a carrier app. While becoming a default app is something the user would allow you to do, only carrier-approved apps could become carrier apps.