Both parties have had stretches where their party chairs have been elected officials. Often, there is a division of labor, where the elected party chair serves as a face-man (or woman, as was the Democrat's Wasserman-Schultz) while a campaign-manager type runs the day-to-day fundraising and candidate recruitment type stuff.
The current buzz in the political press over the DNC race is that Keith Ellison seems to be a good person for the job, but the fact that he's a congressman marks him as a part-timer when folks feel the need for a full-timer. As a relatively-young progressive black Muslim, he would serve as counterpoint to an anti-Muslim, oft-rednecky president-elect Trump. That's too good a package for lots of progressive folks to pass up.
Three guys have their hats in the ring, two of them former governors and presidential candidates (Howard Dean and Martin O'Mally) and one is the current SC party chair, Jaime Harrison. Dean has some progressive chops and helped do some innovation on using the online realm and having a "50-state strategy" that chose not to ignore deep-red states. However, he might be a bit dated for modern Democrats.
One viable option would be to hire someone like Harrison (I know next to nothing about him other than what I read just now), who has experience doing the nuts-and-bolts of party organizing on a state level, as the day-to-day party boss, while having Ellison as the front man who can make the rounds of the Sunday talk shows and be anti-Trump personified. That would give the best of both worlds for the Democrats, still paying attention to the details but maintaining a good speaker.