I believe I’ve just discovered something that sheds light on this mystery. Peter Shore kindly pointed out this vowel chart, in which figure the following two charts (amongst others).

First, the American one:

And now the British one:

This probably explains why it’s so hard for me to find a minimal triple, since General American has only two vowels there, not three. So the reason I could not find a minimal triple is because there cannot be one in General American. We just don’t have that phoneme.

_{Making this answer Community Wiki so that I don’t gain rep from answering my own question.}

I believe I’ve just discovered something that sheds light on this mystery. Peter Shore kindly pointed out this vowel chart, in which figure the following two charts (amongst others).

First, the American one:

And now the British one:

This probably explains why it’s so hard for me to find a minimal triple, since General American has only two vowels there, not three. So the reason I could not find a minimal triple is because there cannot be one in General American. We just don’t have that phoneme.

_{Making this answer Community Wiki so that I don’t gain reps from answering my own question.}