After reading replies to postings from others who experienced more-or-less the same problem, I concluded that Quicktime in OSX10.9.5 has a flaw: when it converts .AVI videos to .MOV, it fails to include the audio.
If you consider anything that doesn't work the way you expect it to a flaw, then yes, the QT X player is flawed. However, it would be more accurate to say that AVI is a legacy file container dropped by Microsoft more than a dozen years ago when Microsoft decided to develop a more modern version of media software. It is also necessary to remember that AVI is a generic file container and not a compression format. Had your AVI files been compressed using MJPEG video and Linear PCM audio, the files would play natively in QT X. On the other hand, an AVI file using DivX 5 video and MP3 audio (another common compression format combination) may have converted without problem by QT X while the same video with DTS audio would have likely lost the audio.
Basically, for a video file to play natively in QT X, the audio and video compression formats must be both supported by the embedded operating system structure and the data must be wrapped file container in which the embedded structure is programmed to recognized the data. When the data is not supported by the embedded structure, the file is sent to the modernizer structure for evaluation. If this structure is programmed to recognize the data format-file container combination and considers it conversion compatible, the the file is passed on to the new conversion embedded structure for processing. It appears that your AVI files are being recognized and being passed on to the conversion structure but it is not clear if you actually have the required additional audio support installed and accessible to the system to actually convert the audio content. And since you have not mentioned what audio format is actually used in the source AVI file, it is not possible to determine if the original audio is supposed to be natively supported or can be supported by the installation of additional codecs.
In the example I gave above, Perian provides conversion support on my system for MP3 audio and DivX 5 video but fails to support conversion of DTS audio even though the system recognizes the DTS playback is supported by the Perian package. In a similar manner, a WMV file with WMA9 audio and WMV8 video can be converted by QT X if the current Flip4Mac codec package is installed on my system even though the source file, like the previous AVI file mentioned is not natively playback compatible with the QT X player. On the other hand, both the AVI and WMV files playback natively in QT 7 with the additional codec support installed. So it may be possible that QT X might be able to support conversion of your AVI files (to include audio) depending on the audio compression format used and whether or not an appropriate codec is available for installation and the modernizer/conversion structures can recognize/is programmed to use it.
Figuring there is no easy fix in MacOS, e.g., by changing settings, I researched apps that can display .AVI videos. I found many, but I wanted to avoid apps that handle .AVI files by importing them into their internal libraries, thereby eating up more storage space on my computer. I just want an app that plays .AVI files directly.
I found one: VLC Media Player. It's a free install. I've tried it and so far it does just what I want.
An "easy fix" depends on whether you wish to fix the problem or work around it and how much effort you are willing to expend. Most common AVI compression formats can be played directly by the classic QT 7 player if the required codecs are installed. In a similar manner, virtually any third-party media player (or converter) based on the FFmpeg framework (VLC, MPlayerX, NicePlayer, HandBrake, etc.) can handle AVI file content and, as you found, provides an easy workaround if you are unwilling to learn more about the actual working of legacy, third-party, and/or proprietary codecs and how they apply to the classic and evolving QT X structures embedded within the various Mac OS X versions.
