This issue, at least for me, was caused by highly specific set of events so this fix might not work in your situation, I only found this fix after scouring though some Activision support forum so I want to make it easier to find for anyone else possibly having similar issues.

I have the physical disc version of Modern Warfare and PS4 Pro. I had installed, played and beaten the campaign mode, but unfortunately I had to get my PS4 sent to get serviced due to system bricking issue. After getting the console back, I re-downloaded Modern Warfare, restored backup system files from USB and when the new season started I thought about going back to catch some singleplayer trophies. Upon launching the game and trying to play any mode, I got an error message telling I was missing DLC packs. Spec ops and Multiplayer tabs threw me to the store page where I could download the packs, but nothing came up when trying to access single-player. Re downloading the game did not work nor did restoring licences.

In the game details page and additional content tab, I could see the singleplayer packs 1 and 2, with a subtext saying they were "disc version" but they had no thumbnails icons and the game file size was too small (around 90Gb, should have been 120Gb ~). The system had not correctly copied and installed the files from the disc like it should have. When I first entered the disc after getting the console back, the disc spun around like it was copying, but there was no progress bar. Only download for a large game version patch.

To fix this, I found recommendation to rebuild the database from safe mode (Note that this is different from restoring factory settings or initializing PS4, this should not delete any data, might delete some folders if you have them set up) . The process took around 2 minutes, I don't have lot of content on the console installed, and after the console booted back up, the disc began spinning and there was progress bar on the Modern Warfare icon and "copying add on" window opened. It took around 20 minutes to copy 30Gb ~ of content. After this the singleplayer mode worked like normal, the game also recognized my old save data which I had backed up.

From what I gathered, the root of this issue (besides the questionable decision to split games into packs which is bound to cause many issues) was me restoring the old PS4 backup data from USB. There was likely some cache data issue that rebuilding database cleared.

This post probably has more words than it needs, but I wanted to add every detail, just in case. TL;DR: rebuild the PS4 databse