The other option is to open the DMG on your old Mac and transfer the items into a freshly created DMG file. Or you may need to transfer those contents first to your new Mac so the new Mac can create the new DMG file for those items.
Apple has made changes to macOS regarding some of their native archives. Seems some people may have trouble opening some older DMGs.....more likely to have a problem if encrypted.
More than likely any third party apps are just leveraging the same low level Apple utilities that Disk Utility & disk image mounter is utilizing. I doubt those utilities would recreate the wheel when they can just wrap an easier to use GUI interface around Apple's own tools perhaps while providing more options.