I replaced the two TL081C and the short calibration works again. The measured values are also quite accurate. Nevertheless, I wanted to calibrate the device according to the "XJW01 LCR Bridge Manual". There are 19 steps. In the steps R3x, G1x and G2x, the corresponding resistors should be used to adjust to 0 pF.
Unfortunately, that doesn't work because the display is not stable. It changes about 3 times per second between small (also negative) pF values up to values of over 100 Henry. No idea if that's supposed to be the case. I don't have any other device to compare it to. The phase shift of -90.04° for the measured capacitor with 185pF (+-1pF) also seems strange to me. The frequencies also deviate from the target values and the peak-to-peak voltage is 400 mV (with 10:1 probe, the scope has not been calibrated for a long time, I also checked the frequencies with a frequency counter). Here, too, I don't know if this is normal.
The clamp diodes are on the other side. The resistor R152 is a bit crooked but the soldering is OK.
Anyway. For my purposes, it is sufficient as it is working now. But I would still be grateful for further information. In particular on calibration and how the software can be read from the STC, as I asked before, so that it can be replaced if necessary.