@erthling The simple answer to your question is, yes, a chromatic has everything that a diatonic has, so it's possible to play almost everything you could play on a diatonic. (Certain long reaches may not be doable.) Whether you should get one depends on what you want to play on it.
Certain things are easier to play on a diatonic, but certain other things are easier on a chromatic. And certain things are simply impossible on a diatonic.
Millions of people play chromatically fretted guitars all the time, and it's not the least bit controversial. Electric guitars, 7-string guitars, and harp guitars are not at all controversial, and I've never heard anybody ask if they're something other than a guitar. Guitars play everything from simple melodies to strummed cowboy songs, to classical etudes, to rich jazz chords, to fast bebop and bluegrass solos, to Stanley Jordan's two-hand style, to Van Halen-style shredding; while you or I may find some styles uninteresting or ugly, nobody says it shouldn't be played on the guitar. Clearly, guitars have progressed unimaginably far beyond their traditional roots.
While respecting and loving dulcimers' traditional, diatonic roots, there's no reason why the instrument needs to be limited to that.
If I think a song would sound good on a dulcimer, I use the instrument that would be best for the job. Left to my own devices, I generally play dulcimers with 0+/1+/6+/8+/13+ frets and fully chromatic ones, because they generally do what I need. I rarely play my diatonic, except for teaching purposes. And I fully respect the right of others to choose differently.
I know some people question whether or not a chromatic should be called a dulcimer, but the instrument doesn't care what you call it.
So, again, whether you should get a chromatic depends on what you want to do with it. It's totally okay to get one, and it's totally okay not to.
PS - One option is to get an inexpensive cardboard chromatic, and see how you like it.