As I recall, they aired them mostly between features like other shorts on AMC, such as Paramount/Jerry Fairbanks Speaking of Animals, Popular Science and Unusual Occupations... all shorts that SHOULD be shown on TCM today. These Soundies were like shorter (mostly one or two songs) variations of Warner Brothers' Melody Masters (running a full 10 minutes with up to six songs and sometimes a "plot") that are shown on TCM a lot. Universal made even longer jazzy reels, running 15+ minutes. Of course, you can search in each of the Shortie Checklists here for those titles and musical performers.
TCM should have no issue showing these since they are also movies. Some were shown in theaters as well as jukebox joints. TCM also shows many educational 16mm films marketed for schools and businesses.
I think the Soundie that gets shown the most on TV and is included in some of Kino's DVDs of Paramount shorts (even though Paramount didn't produce it) is the one with Fats Waller's "Ain't Misbehaving".
If TCM had more motivation to show non-MGM and Warner shorts (aside the occasional "public domain" or mid-50s RKO), there is a huge treasure trove of material available and begging for an audience and a $$ market. Just imagine if Universal, Paramount, Sony/Columbia and Fox suddenly discovered that people here would be willing to buy DVDs or online downloads of these.