Springfield Armory M1903's with serial numbers generally under 800,000 were all manufactured using a technique described as a Single Heat Treatment (SHT). (NB: The same technique was used at Rock Island with a slightly higher incidence of problems but of course with a separate range of serial numbers.) While the basic SHT process was adequate in a small percentage of cases the receiver steel was "burned". This resulted in a receiver that was outwardly OK and could very well provide good service for years until it 1. spontaneously let go or 2. some other factor such as a bad cartridge case released gas into the reciever ring; the wrong ammunition was used (8mm Mauser in several cases).
During WW1 Ordnance addressed the problem by changing the manufacturing process to a Double Heat Treatment (DHT) process which resulted in a receiver with a substantially higher margin of safety. These changes were introduced about s/n 800,000 and is what accounts for the general guidance cited in your original post. Ordnance also went to private industry and fulfilled most of the WW1 rifle requirements with the M1917 (which used a nickle steel receiver).
During the 20's Ordnance studied the issue of the low numbered receivers in detail. Keep in mind every rifle that made it to the field had been subjected to an over pressure proof test followed by a function test. I don't know what percentage of low numbered rifles were destroyed in testing. But reading between the lines in Hatcher presumably some of them were. In the final analysis Ordnance concluded there was no pracitical or cost effective way to 1. Identify the rifles with "burned" receivers and 2. re-heat treat low numbered rifles generally. So the decision was made to retain the low numbered rifles as a war reserve and scrap low numbered receivers when the rifles came in for repair or rebuild. From that point the so called low-numbered rifles (SHT) were lumped together and considered unsafet to shoot.
If you are interested in more detail on the subject "Hatcher's Notebook" by Major General Julian Hatcher, Stackpole is an excellent source as the author was personnaly involved in many of the issues. There is pertinent information in several different sections of the book including summaries of all of the accident reports up to 1929.
I hope this helps. Its kind of a thumbnail explanation and there is a lot more to the story.
As was mentioned earlier somefolks will shoot and have shot their low-numbered rifles with out problems. Their rifles may be Ok or they may have been lucky and never had a cartridge case let go, get a few too many grains of powder in a reload or what ever. When it comes to giving advice to someone else most of us err on the side of caution and recommend against shooting these veterans.
Regards,
Jim