Jane-DoH:
On the J57 as you (OldBUFFkeeper) describe, they were to deal with the fact that the LP compressor was drawing in more air than the HP stages could handle, so they put a bleed-valve in to get rid of the excess air until the RPM increased passed a certain point.
True, but more to the point:
After high power ops, when throttled back, the LP compressor continues to attempt to pump air into HP stages that can no longer handle all that mass flow, and the LP is thus back-pressured to the point of airfoils stalling i.e. surge. The bleed valve opens to dump the excess flow and thus permit a smooth decel.
P&W and (I believe) R-R used this method from the 1950s onward. GE found it necessary to employ a similar scheme beginning with the CF6-50 in the early 70s, although the GE system dumps into the fan discharge duct.