Big difference in operation between OBD1 E36 M42 and OBD2 E36 M44? which are we talking about? E36 M42 can run fine without O2 sensor but will be on the rich side around 13.5 while cruising .
E36 m44 I can say 100% because I own one. Other I do not know. But in any case even for E30 m42 M1.7 they say: DME performs the folowing operations - adaptive oxygen sensor regulation. It means DME fine tune mixture regarding oxygen sensor measurements. Some guys say there is no difference with or without lambda, but probably you will not feel small difference sometimes and depends how close fuel mixture is from optimal mixture with lambda adaptations.
I am not DME guru but in all cases my cars (E30 m42 and E36 m44) was running much better with lambda sensor. In E30 case I had no possibility to read DME data so just pedal test was done by me.
Quite often lambda sensors are damaged during installation because thread was rusted and by forcing new lambda in to rusted thread you just make ceramics inside to crack. You must clean thread and tighten lambda just few Nm avoiding overtorque.