Half my tracks disappear on my Mac after syncing music with iPhone or iPad.

The "missing file" issue with exclamation marks happens if the file is no longer where iTunes or Music expects to find it. Possible causes are that you or some third party tool has moved, renamed or deleted the file, one of its parent folders, the drive it lives on has had a name change, or you've moved a non-portable library to a different path (see Make a split library portable for details). It is also possible that iTunes or Music have changed from expecting the files to be in the pre-iTunes 9 layout to post-iTunes 9 layout, or vice-versa, and so is looking in slightly the wrong place, (see the iTunes Media Organization section of Managing your Mac media libraries - Apple Community for details) or that you've been too aggressive when deleting duplicates at some point.

Select a track with an exclamation mark, use Cmd-I to get Song Info, then click No when asked to try to locate the track. Look on the file tab for the location the library thinks the file should be. Now take a look around your hard drives. Hopefully you can locate the track in question. If a section of your library has simply been moved, a folder renamed, or a drive label has changed, it should be possible to reverse the actions. If the difference between the two paths is an additional Music folder in one path then this is a layout issue. I can explain further if that is the case. If everything is where it is supposed to be try Repair security permissions for iTunes for Mac - Apple Community.

In some cases the library may be able to repair itself if you go through the same steps with Get Info, or when playing a track, but this time click Locate and browse to the lost track. It may then offer to attempt to automatically fix other broken links. Although it says something like "use the same location" I think it expects to find the tracks in the same artist & album layout they were in previously, with one systematic change to the path.

If you want me to try to provide specific advice please post back the following details:

1. The location of the media folder under iTunes|Music > Preferences > Advanced
2. The location of a sample missing track shown under Song Info > File > Location that begins file://
3. The true path to the file whose details you gave in 2

See also FixLinks - an AppleScript to repair broken links in Music - Apple Community.

tt2

Syncing your device shouldn't remove locally stored files. I cannot think of a plausible mechanism for this. Nevertheless you ought to keep a backup of your library such that it could be restored in the event of something going wrong. See Back up your Mac with Time Machine - Apple Support.

tt2