Why is OTA DTV audio easily broken with weak signal?

FM signals are more robust than AM, which explains why NTSC sound is sometimes present when the picture is unwatchable. ATSC sound cuts out when the picture is slightly garbled because there is a loss of information that cannot be corrected and decoded. Dropouts in the audio is more annoying than a slight loss of picture information. In the early days of ATSC (and other forms of digital HD), the picture would go black when there was a video dropout. That was very annoying. Today's video decoders buffer the last frame so that video dropouts are less noticeable. Unfortunately, there is no counterpart for buffering audio so audio dropouts are as annoying as ever. There may be other reasons why audio dropouts are more noticeable, such as encoding/decoding methods, but that's my take on it.

seriously robust error correction

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Return signal or not, high levels of error correction require high amounts of redundancy, lots of overhead and time. For example, deep space probes send pictures hundreds of millions of miles but require days to send one picture frame. That's not a good option for ATSC TV. Error correction is always a trade off and will always fail under conditions it was not designed to handle. I'm sure the ATSC standards committee took all factors into consideration and came up with an acceptable trade off for what is required with terrestrial TV broadcasts.