Is it possible for me not to fill the holes, but still fix the typology using the remesh modifier?
Yes, but it’s really tricky and will need some manual editing in the end for a result that may not be really satisfying :
-In Edge select mode, select the edges of the holes, and extrude them down to give those holes a “depth” :
-Use the Remesh Modifier at a level of Octree that looks good for you and preserve your model shape
Now delete manually the extra faces in those holes (go into Face select mode, and press ALT + Right Click next to an edge of one of the face you want to deleted to select at once a whole loop of face, making the face selection for deleting quicker)
The main problem after that is for you to decide if the mess you may obtain around those holes with the remesh modifier better than the mess you had originally ?
Without even mentionning that the face count will be higher, because with the remesh modifier to keep your details you may need a high octree (and most of the time leading into more faces than your original model had)