IMO, the '03 is not the masterpiece many think it is. However, the A3 verson fixes its worst trait, the rear sight. An '03 A3 should serve you nicely if it's not overly worn.
The Lack of the complete story and Hollywood are the reasons people love it. Its competition, the US version of the British Pattern 1914 known as the "US Rifle, Model of 1917" is a superior rifle. Some people call this rifle the "US Enfield", but Enfield had little to do with it, as 100% of all British Pattern 1914 production was done in the US by contractors. This is the same reason it didn't get selected as the US standard rifle post WWI. The Government owned the arsenals producing the '03. The '17 was a contracted item at a time when labor disputes commonly interfered with production. The primary reason the '03 beat it was complete control of production.
Bias note: In my short collecting career, I've met (and read) dudes who would put a piece of crap under glass, if it was shit by anyone working in the Springfield arsenal.