The speed sensor is monitored in at least a couple different ways. It is monitored for internal resistance/resistance to ground and its output as a signal generator. In the second activity the various readings for other sensors are used to compare readings of "mapped" values. If the car is going straight (as determined by three confirming wheel speed signals), then a fourth disagreeing value will be faulted (in ASR 4 sensor systems). If it is in a right turn the left front sensor will be traveling the fastest and the right rear the slowest.
In a article I wrote about BMW transmissions there is a scope pattern of the digetized speed signals coming to the trans controller. There are two 4 channel patterns at the end of the article, one in straight driving and the other in a hard turn:
http://www.continentalimports.com/ser_ic100076.html
The first type of failure is seen on startup during the selftest. The second type of failure is performance based and occurs once 10kph is reached and is continuous. In your situation the type of failure has changed and there isn't any direct reason. You now have something that isn't passing the selftest. The sensor can only fail the selftest due to resistance problems. Since you now have different type of failure it could be for most anything. The OVP only powers the ABS controller, not the light. It takes a manifest decision by the controller to turn out the light. The light will be on if the controller is removed from the car.
One point I might make about purchasing diagnostics: Almost all diagnostic fault trees leave the technician at this point: try known good device and retest. If one is buying diagnostics by the drop so that one can then purchase said component more cheaply, then one has become part of the diagnostic process.
I do regular testing for numerous other shops. They all wish to make the money on parts from my diagnostic. I let them buy as much of my testing and intelligent speculation as they wish, BUT if they wish a finished product with a warrantee I provide the parts.
LET ME TELL YOU WHY. Diagnostics done well is a bunch of speculation. It needs to be that way, this is not scientific research and recieves no government spending. Using, seat of the pants statistics, thorough knowledge, and all the testing technique (known to mankind) a good diagnostician can determine that the ABS controller is probably the problem (say for instance). There is no way to prove it except substitution. If that tech is so equipted with a ready substitute then diagnostics is easily finished. If NOT then either the shop has to warrant the diagnosis or the customer has to (by accepting parts that don't fix problems). The state of diagnostic risk described here is maximum for control units which can't be tested in the field, but all diagnostics is a work in progress till the final repair is finished. The level of testing is thus left to the one extending warrantee. If left in the hands of an honest professional one will buy only the parts needed and leave with a warrantee.