I finally got my app update through App Review without any changes to the update.
TLDR: After months of communicating with my app reviewer as well as with a sympathetic tech support representative, they eventually approved my update without any apparent fixes to the ODR backend.
On Monday June 1, I submitted the same build for review again, and it was rejected the same day. I replied to App Review via the "Resolution Center" that I intended to follow the instructions from tech support (see my post above) and to submit the build everyday until the issue with their backend was resolved. I then received a completely infuriating response: "We appreciate your efforts to comply with the App Store Review Guidelines, and we look forward to reviewing your resubmitted app." I replied, expressing my frustration with the seemingly canned, robotic responses and pointing out that this issue had nothing to my "complying with the App Store Review Guidelines." They responded by setting up another phone call "within 3 to 5 business days." The reviewer (same that I had spoken to in late April) called me on Friday June 5 and explained that he had no visibility into Rich's progress on the issue and promised to track down engineering and follow up on my bug report and tech support incident. On Monday June 8, I received the first response to my Feedback Assistant bug report: another completely infuriating, inhuman instruction "Once the issue has been reproduced please trigger a sysdiagnose after the problem occurs and attach the log file to Feedback Assistant." I replied to that message, as well as to my app reviewer via "Resolution Center" and the tech support incident email thread, that there was no way for me to reproduce the error -- in fact, this was at the heart of the entire issue! A few hours later I got a call from the same app reviewer: he said this was extremely unusual but they were going to approve my app update without a fix! He offered an apology for my trouble but didn't explain what the underlying issue was. My update indeed went through and is now live with the ODR requests working as expected.
My only suggestion is to be persistent with complaints to App Review and to Developer Technical Support until they either fix their backend problems or approve your app update as is. It may be helpful to reference my reports:
Developer Technical Support Incident Case ID: 732675555
Feedback Assistant bug report ID: FB7655668