Replace the laser pickup.
The pickup is actually "2-in-1", i.e. there is a CD- and a DVD-part. If one of the two fail, the whole pickup needs to be replaced.
Unfortunately, the replacement pickup (even on Aliexpress) is quite expensive. (40-50 USD, but look around, maybe it's cheaper somewhere)
I dismantled the drive today and the pickup is (as stated in the service manual) a KHM-220AAA.
I see what you mean about the laser being a 2-in-1. I have watched a whole bunch of YouTube videos and most of the pickups out there have two separate lasers with two adjustment potentiometers. I can see that this one only has one potentiometer, so as you correctly pointed out, this indicates the 2-in-1 laser and because of this I cannot adjust the DVD pickup separately.
That being said, I tried to adjust the potentiometer anyway (you never know).
To recap, the player plays CDs fine but not DVDs apart from sometimes.The initial value was 1.461KΩ I adjusted it as follows:
1.098KΩ - Made no difference
0.791KΩ - Made no difference
0.341KΩ - Made no difference
So I decided to adjust it the other way:
1.996KΩ - Sort of read a DVD. Would load the initial language select screen but when trying to load the menu it would not track and then say error C13
2.38KΩ - Same as the last reading
2.63KΩ - A little better this time. Picked up the disc straight away. The DVD loads the menu but will not play any videos (main feature or extras).
3.12KΩ - This would not work at all. Could not detect DVDs or CDs at all.
4.64KΩ - Same as above.
So I have adjusted it back to 1.465KΩ (as close as I could get back to original. And now the problem is back to CDs working but not DVDs.
I tested 4 different DVDs (2 Single Layer and 2 Dual and also 2 different CDs with every one of the above readings.
So This is probably the pickup like you said as far as I can tell.
I have emailed Sony to see if they sell the pickup still. They will probably charge a fortune though. The cheapest I can find one is £28 on Ali Express UK.