They have to have the juice to start the APU, and then the charging system carries the load. Once the APU starts, the Batteries are done. If the Batteries are charged by the APU, then obviously the APU is powering the system, not the Batteries.
While simple and atractive as a concept I suspect that restricting battery charging to the ground would have ripple effects.
For example what % of the APU battery charge is used by a single start cycle? How long does it take to "top off" after that and how many start attempts must be avaialable in non-nominal flight?
While these factors might have been fairly easily accounted for if "ground charge only" was in play early on, retrofitting it would probably take a signifiicant amount of effort to revisit all the analysis that assumed fully charge batteries, even assuming the basic capacities where adequate.
Of course this also assume that the actual fault involves the charging system rather than an internal battery fault.