Mind absolute maximum ratings; they don't want you to apply more than 2V beyond the rails to the inputs.
OPA627 has multiple paths from Vs- to Vs+ (
my schematic here). I quickly found:
- Q25, Q22, Q12, Q15, Q32, Q33||R24
- Q9, Q5, Q13, Q32, Q33||R24
Needless to say, voltage drop may be quite dependent on test current.
For very low test currents, it could possibly be near 2V. If your DMM diode range also tops out at 2V, your conclusion about the real one being open circuit may be incorrect.
All opamps produced on oldschool, junction isolated, noncomplementary bipolar process with PNP or P-JFET inputs necessarily have diodes from Vs- to each input. This applies to TL072 and friends, LFxxx, AD744 by the looks of it, LM358, RC4558, others. Diodes from the inputs to Vs+ may or may not exist. CMOS tends to have diodes both ways because gate dielectric is very sensitive to ESD of either polarity.
Your TL081 may still be LM358 - test its offset pins

I don't know of any old PNP input single opamps with offset trim, though it doesn't necessarily mean that none existed.
If there are no diodes from Vs- to the input pins it may still be NPN input or a complementary process or both. But those input stages are equipped with differential protection: antiparallel diodes between the pins (NE5532) or pairs of diodes (LT1028), sometimes with series resistors (OP07).
edit
I forgot about LM101 and µA741 style input stages. The former has an obvious parasitic diode to VCC but I suspect that both can stand several volts below ground. No differential clamping.
More reverse paths found through OPA627:
- Q30, Q35, Q32, Q33||R24
- Q9, Q7, R5, R7
- Q10, Q8, R6, R8
The resistances involved are unknown, but a 2V drop isn't entirely impossible.