BP specifically means that the date was calibrated [1] after radiocarbon dating. It is not just an epoch difference.
Also, the radiocarbon dating itself (and thus the beginning of effective absolute dating) was invented around 1950, and nuclear weapon tests made everything after that point unsuitable for the radiocarbon dating. It would be natural to have that point as a reference for the "present" at least for now.
I actually never thought about the fact that human nuclear testing would basically destroy the accuracy of radiocarbon dating. Any distant future civilizations who assumed there had never been any ancient sources of distortion would end up with results that are way off. Of course, that’s true of us as well.
Technically we are still able to use radiocarbon dating for dates after 1950, at the reduced accuracy though. This is what I meant by "unusable", because other dating techniques will work better.
The calibration is there because the global 14C/12C ratio is not constant over the time, and the same procedure can be used given the recent history of ratios (that we do have). As far as I know, though, resulting error bars might be much larger than usual because the function from uncalibrated ("14C") years to calendar years is no more regular. For example see [1] where uncalibrated t2 and t3 can correspond to much larger periods than t1's; historically this happened only locally, but after 1950 it would be far more common. Still the radiocarbon dating can be used with this caveat [2].
Also, the radiocarbon dating itself (and thus the beginning of effective absolute dating) was invented around 1950, and nuclear weapon tests made everything after that point unsuitable for the radiocarbon dating. It would be natural to have that point as a reference for the "present" at least for now.
[1] https://en.wikipedia.org/wiki/Radiocarbon_calibration