The 1903 and 1903-A3 Springfield actions, both have a magazine cut-off.
This was so they could function as a 'single-shot', with a cartridge loaded directly into the chamber and the magazine contents held in reserve, until rapid magazine feed was needed.
The 1903 and 1903-A3 extractors are shaped to 'snap' over the base of a chambered round.
Rifles will vary in how smooth this happens.
It is likely your extractor will smooth up with use, or a little bit of 'stone work' to remove any burrs or surface roughness.
FWIW - I have installed several NOS Springfield extractors on S.A. 1903 and Remington '03-A3 bolts. It was easy and the parts worked well.
Out of habit, when 'single firing' bolt-guns with rimless cartridges, I invariably push the round down on the follower, (with magazine-empty), and feed the cartridge from the magazine. It makes for a smooth feed and less flexing of the extractor. (Rimmed cartridges, like the .30-40 Krag, I feed directly into the chamber. With Lee-Enfields, I press the cartridge down into the magazine).
I did put a couple of Remington marked extractors on a Remington Bolts and likely Springfield Armory extractors on S.A. Bolts. No fitting was necessary.
At around $15, a spare extractor is a useful part to have.