Adding multiple differential signal
... we can only use op-amp, instead of lovely integrated circuit (ough).


An op-amp is nowadays almost always an integrated circuit. Be glad you don't have to make them out of transistors!



Figure 1. A summing differential amplifier using only one op-amp. Source Section G10: Differential Summing by UNCC.

The article linked above discusses the maths in some detail. I haven't read it through but I suspect that you need to be aware of limitations of the circuit due to possible degradation of common-mode rejection ratio (CMMR) or offsets caused by bias currents.