Good points about starting with songs you already know well. But I have a different perspective:

The songs you already know are not necessarily the best choices for concertina accompaniment. Either they are songs that sound perfectly complete unaccompanied, or they're songs that probably better suit the other instruments you already play them on. Obviously, there are exceptions - songs that translate well, and may even be improved with the box. But it doesn't take long to learn to sing a new song comfortably - just one verse is all you need to start.

There's also the matter of motivation: One of the most tedious, unsatisfying things about learning a new instrument is having to rehash old chestnuts with simplistic accompaniments. The songs I first learned on the box were ones I didn't know. They sounded so right with concertina, I was eager to learn them, and I was undaunted by their initial difficulty. By emulating the best, I also quickly picked up on what works and what doesn't.

I'm sure Dick's tutor provides an excellent progressive grounding, using both familiar and lesser-known material. After mastering that, it probably doesn't matter whether you revisit old friends or tackle new songs, though I find the latter more motivating - and suitable.