Did... rep is looking into it but says EMCopy has "no support path".  Confirmed we are using the latest.  He replied that it appears to be working as designed but to be honest, this doesn't make too much sense to me and I'm thinking he is misunderstanding the root issue (seems to be only affecting removing directories on the second pass, not files at any point, when permissions are set so that the admin doing the copy is normally not permitted to access that directory).  I admit, it's a pretty specific circumstance when the issue comes up, but it's unfortunate since the EMCopy tool is so useful in other cases.

Here's what he replied in case I'm the one misunderstanding his position.  I don't have access to the link he provided, just going off his summary that he included:

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I found the following article here
https://emcservice--c.visualforce.com/apex/KB_BreakFix_clone?id=kA2j0000000QiJb

The article states the following, which I believe fits your criteria regarding permissions to the directory.
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From version 4.04 onward, stricter checks have been introduced to EMCOPY in order to prevent copying unstable files, for example, files that are being modified at the time of copying. The reason for introducing this change is that copying files that are in the process of being edited may lead to an error during copy (best case) or to incorrect file content (worst case) without any notification to the operator. EMCOPY will retry the copy, depending on the EMCOPY parameters used. If all copy retries fail as well, EMCOPY will try to delete the destination file in order to not leave any partially copied file on the destination.
EMCOPY will only try to delete a file on the destination if it encountered an error during the copy that led to an incomplete copy, that is, an invalid copy on the destination. If this deletion succeeds, a new differential copy can be run to copy any file across that couldn't be copied previously. See 71643 for details.
The message "DELETION FAILED" shown above means that EMCOPY could not delete the destination file after it failed to fully copy it. Reasons for this are environmental, for example, because another process is using the same file, the destination is no longer reachable, EMCOPY is not run with the correct permissions.
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So it appears that it is working as designed. This does feel like a potential request for enhancement from your end. I have been unsuccessful in finding a support path for EMCopy, but if you want me to continue to pursue that I can.

Otherwise, the current version of EMCopy you are running is the most current I can find, so you seem up to date on that.

Make an SR since EMCopy is supported.