Two questions really. And maybe the answer is so long and involved that it's too much to ask for an answer here, but anyway....

In the music and songs I've seen, the fully diminished, non-diatonic, dim chord is way more popular than the half-diminished chord. Maybe it's genre specific (I don't know about jazz), but it seems to stand alone as a chord that is more popular as an altered chord than the basic diatonic chord. Is the fully diminished chord just especially useful, and/or is the half-diminished chord not very useful musically...or something else?

Related to this, the diminished chord also stands alone as a chord where the 7th form (either full or half-diminished) is the norm. I don't think I've ever seen a chord chart where a basic diminished triad (e.g., BDF) is called for. Not saying it doesn't happen, but in my limited experience it's always with the 7th. Again, not sure why that is.