They can. I've heard of cases where the brushes on the disk knocked the lens assembly out of alignment, completely ruining the drive.
Also, if there's a layer of persistent dust or other gunk built up on the lens, then the brushes on the cleaning disk may abrade the lens surface when brushing it off. Scratches like this cannot be repaired, and once again you're likely to have a ruined drive if this happens.
Some forms of lens contamination simply can't be removed by a cleaning disk. For example, if anyone has ever smoked (tobacco or cannabis or ...) in your car, then there's a good chance that the lens has been coated with smoke particles and tar. Tobacco-smoke tar is extremely difficult to remove, even with proper cleaning fluids (the Consumer Reports magazine uses it as its exemplar of a "hard to remove" surface film). I have no reason to believe that cannabis tar is any less damaging to an optical drive.
Entirely possible.
Equally importantly: imaging the DVD in a computer drive is a non- destructive operation. It won't alter the DVD or drive in any way. If it works, great - burn yourself a copy on a high-quality DVD-R blank and see if it works in the car. If not, you haven't lost anything other than time.
Unfortunately, I think that other people have likely pointed you in the direction of the real cause: the DVD drive mechanism (likely the laser assembly) in the car is failing due to old age. If this is the case, replacement is the only real solution.
Frankly I'd suggest a different solution. A cheap, outdated cell-phone, running Android, with a navigation package like OSMand, will provide you with the same functionality (maybe better), with maps that are updated on a regular basis, and no moving-parts mechanism to fail. OSMand is free to try out (unlimited time of use, but only a fixed number of map downloads or updates available) and cheap to buy (possibly the best $5 I have ever spent).