To clarify for the folks that are saying it's not glitched for them, this problem is specially referring to the case in which you finished the DLC on PS4 and are now attempting to do the same on PS5.
The actual steps that lead to this issue are something akin to:
1. On your PS4, complete all 4 DLC missions and unlock the trophies.
2. On your PS5, start up Outriders.
3. As you've already completed all of the missions on your PS4 and it's saved on the Server, the trophies should automatically unlock.
4. However, you'll see that the missions are listed at 0/4 or 3/4 and no matter how many times you complete them that counter won't go up.
If you started and finished the DLC missions for the first time on PS5, then you won't encounter this.
edit: As a side note, I believe the reverse order does work. So if you completed it on PS5 first and then go to PS4, the trophies should automatically unlock as expected -- it's just PS4 to PS5 that's having problems.