"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 don't actually know what Migration Assistant does as I've never used it. However, based on the rsync example below, I've shown it will preserve your timestamps (and other metadata you may have lost).
I used rsync ver 3.2.4 for this answer, installed from MacPorts. I can't guess what features your version of rsync includes. You should consider upgrading rsync with MacPorts or Homebrew. Apple hasn't upgraded "their" version of rsync in many years, and it lacks the ability to support the methods and options used here.
Background: I began using rsync as my primary backup tool for macOS some time ago because I had to address the same issues as you: mangled and missing metadata. In my case, the issue was perhaps exacerbated by crossing a file system boundary - from APFS on my local drive to btrfs via SMB on a 'Synology NAS'. In your case, using Migration Assistant, I cannot imagine how Apple could not get the timestamp metadata correct - but you're not the only one who's reported this issue.
I made some notes while working to resolve my issue, and they morphed into a "recipe" in this GitHub repo. I'll keep this answer brief as most of the details are covered in that document.
Q: Can I use rsync to fix the timestamps that Migration Assistant trashed?
A: Yes - as shown below and verified
rsync with metadata preservation
rsync effectively copies data from a source (the "from" location) to a destination (the "to" location). If you are using a fileserver (NAS) as a backup for your Mac, the GitHub recipe will provide complete guidance to make and restore an rsync backup without losing metadata (attrs & xattrs).
However, as you need only to "migrate" files & folders from an old Mac to a new Mac, there is no need to involve an intermediate NAS. Consequently, the GitHub recipe can be simplified.
The scenario below assumes that the remote folders have been mounted (perhaps via 'File Sharing' in 'System Settings'). Alternatively, if you prefer to enable 'Remote Login' you can run rsync via SSH.
The following command code block shows the command syntax & options needed for rsync to perform an old Mac-to-new Mac "migration":
SCENARIO: from "oldMac" to "newMac"
Assuming the source & destination folders are:
SOURCE: /Users/MyHome/oldMacFolder/
DESTINATION: /Users/MyHome/newMacFolder
From Terminal.app on your "old Mac":
% SRC-FLDR="/Users/MyHome/oldMacFolder/"
% DST-FLDR="/Users/MyHome/newMacFolder"
% rsync -rlAXtgoDivv --dry-run -M--fake-super $SRC-FLDR $DST-FLDR
Notes:
--dry-runthis option preventsrsyncfrom actually copying any files! It's used so that you can see what will be copied once the option is removed.If you've not set up file-sharing to mount the remote folder(s), you may use an
SSHconnection by adding the option-e sshto thersynccommand. This will require that you've set up 'Remote Login' in 'System Settings'.
Verifying Results - Preservation of All Metadata:
The GitHub recipe includes a short zsh script that will stat all of the files, folders, links, etc between the SOURCE and DESTINATION folders. It may be run after the rsync operations to verify that the chosen stat attributes have been preserved.
The rsync options presented here will do more than preserve the metadata containing the date-time stamps; they will also preserve all extended attributes (xattrs) . The GitHub recipe also includes a simple script to compare the output of xattr -lrsvx for all files (that have xattrs) in the SOURCE and DESTINATION directories.
The stat utility that I use here is the GNU coreutils version 9.1. The find version I use is the GNU findutils version 4.9.0. Both are readily available through MacPorts (and likely through Homebrew) for virtually every version of macOS. The xattr command is a macOS utility.