[BRX]

Well, Trevor, I think you see the difficulties yourself if you ask "how a program can be clever enough". 

For PDFs there simply isn't a good transposing method. Even the OCR programs for note recognition aren't quite there that.
(Scanscore seems promising, but it's not cheap and they changed to a suscription mode with the new version which I dislike).

You can look into Band-in-a-Box too. That's practically a very sophisticated and mighty "iRealPro for PC" which also can display
melodies (and is actually much longer on the market).

It's also not cheap. But you will be able to find files and songs for BIAB with melodies and chords for a lot of standards (not text though) 
which you can transpose and print out or save as a pdf to display on your phone.

If you are willing to invest time and work to transcribe a PDF into transposable electronic form MuseScore (or any other notation software)
is indeed the way.

I'm very partial to the aforementioned ABC though which is an ASCII notation language which is very simple to learn (at least for the
basic stuff). And if you just want chords/melody/text you can very quickly write down an ABC file which can be displayed and
printed in a lot of (free) ways. I won't dive into that further. But I have quite often transcribed songs from real books into ABC
to be able to transpose them on the fly or quickly edit something and display or print it.

But actually, if you won't use a tablet but only your phone IMHO you should stick to iRealPro, because the melodies probably are
displayed so tiny that they aren't to useful (but maybe you've got very good eyes).