The offset voltage should be no longer a big problem, as the modern system tend to use rather high resolution ADCs and can live with some offset. Besides offset there is often also some hum as background. The background from hum and offsets limits on how much gain can be used, but with a good ADC there is enough dynamic range and not that much gain needed. If a 16 bit ADC works with LSB steps of some 100 nV for the input before the amplifier, it can still read signals up to some +-3.2 mV. So the DC offset and hum only need to be smaller than some 2 mV_peak for this example.
In the old days the likely used more analog filtering (e.g. 50 Hz notch), offset trim and maybe AC coupling with a low cross over, to get away with less ADC resolution or direct plot to paper.
I have the feeling that the signals are in many cases not actually limited by the amplifier or ADC noise, but some background from EMI, hum and persons moving / breathing. So there may not be a need to have the absolute lowest noise amplifiers. In many cases the designs also care about power to allow battery operation and this way get good isolation.
It may be a bit more critical with small animals in a well shielded lab environment, compared to a more normal hospital environment.
Edit: with more critical, I mean less background and thus a possible use for a lower noise amplifier.