"I have discovered the hard way that Apple's "Migration Assistant" doesn't always preserve file timestamps"
comment: It seems that has become Apple's MO - you must discover all things the hard way.
Caveats: I'm not entirely sure this will resolve your issue or answer your question, but I hope it will help. I don't actually know what Migration Assistant does because in 12 years and 5 different Macs, I have never used it. So this answer may be a partial answer, or it may be incompatible with Migration Assistant. As a guess, I'd say use rsync after Migration Assistant - to clean up the mess it has left. Others here may have better ideas; the rest of my answer will be limited to rsync.
The rsync version I used was installed from MacPorts; version 3.2.4. I do not know if the rsync version included with your version of macOS includes the required options or not - you may wish to investigate that instead of installing a current version of rsync.
Background: I've begun using rsync as my primary backup tool for macOS recently, and had to address some of the same issues as you: mangled and missing metadata. In my case the issue was caused by crossing a file system boundary - from APFS on my local drive to btrfs via SMB on a Synology NAS. In your case - Migration Assistant - I cannot even imagine how it could not even get the timestamp metadata correct, but you're certainly not the only one who's reported such issues.
I made some notes while working to resolve my issue, and they morphed into a procedure on my macOS GitHub repo. I'll try to keep this answer rather brief since most of the details are covered in that document:
rsync with metadata preservation
rsync effectively copies data from a source (the "from" location) to a destination. If you are using a fileserver (NAS) as an intermediary between your two Macs, these commands may do what you need:
from "old Mac" to "NAS":
Assuming the source & destination folders are:
SOURCE: /Users/MyHome/MyData/
DESTINATION: /System/Volumes/Data/mnt/MyNAS/
From a terminal on your "old Mac":
% SRC-FLDR="/Users/MyHome/MyData/"
% DST-FLDR="/System/Volumes/Data/mnt/MyNAS/"
% rsync -rlAXtgoDivv --dry-run --fake-super $SRC-FLDR $DST-FLDR > rsync.log 2>&1
Explanation: This rsync command uses the --dry-run option, and therefore will not actually move or modify any files. It will give you a detailed log (the ivv options) of what files would have been moved or modified. Once you are happy with the results, simply remove the --dry-run option, and run again. Review the rsync.log file using the "Decoding Table"
from NAS to "new Mac":
SOURCE: /System/Volumes/Data/mnt/MyNAS/
DESTINATION: /Users/MyNewMacHome/MyData/
From a terminal on your "new Mac":
% SRC-FLDR="/System/Volumes/Data/mnt/MyNAS/"
% DST-FLDR="/Users/MyNewMacHome/MyData/"
% rsync -rlAXtgoDivv --dry-run --fake-super $SRC-FLDR $DST-FLDR > rsync.log 2>&1
from "old Mac" to "new Mac"
If you can make an SSH connection between your "old Mac" & "new Mac", I see no reason to involve an intermediate NAS. The same rsync options may be used; I'll leave it to you to formulate the SOURCE and DESTINATION folders.
Verifying Results:
In the GitHub recipe is included a short zsh script that will stat all of the files, folders, links, etc between the SOURCE and DESTINATION folders. It may be run following rsync operations to verify that at least the stat attributes are the same.