The first thing I quickly note is the 'straight 'bolt handle and the restamped ser#.
The straight bolt handle was a trait of the very early production '03s,,the low# rifles.

The restamped ser# throws a red flag up for me on a 1903 that the ser# you now see on it could be misleading in that it could have been purposely stamped as a 'high number' to take the rifle out of the less desireable 'low#' catagory.

The old style straight bolt also being a clue to a low# 1903 may have been left in the rifle by an uninformed conversion wizard.

There are probably other minutia to tell the age or era of the recv'r that I'm not aware of.
I'm not saying it IS a low# recv'r, but I'd at least suspect it and investigate further.

The low# recv'rs are very hard and to engrave them (or drill/tap them) you have to break through the case hardened surface.
Carbide engraving tools will sometimes do that for you but the smoothness of the cuts won't be there.
Another way is to polish through the 'case' to get to the softer core metal. A lot of work.
The usual way is to anneal or at least draw back the hardness to a degree so you can cut it.
That would include both the rec'vr and the bolt body here if that was in fact the case.

The nickle steel actions cut pretty well as they are.
The double HT actions are still hard on the surface as the Low#rifles but sometimes can be cut through with little problem,,some not.

The reddish cast of the re-blue on the rec'vr and the bolt body (handle) is another hint in that direction IMO,,common to casehardened parts.
...But also could be the result of other re-bluing problems and could show regardless of steel type and HT.
If the latter, I'd expect the red color to also show up on all the other metal reblued parts assuming the projects parts were colored at the same time which would be a most common occurence if depleted salts, poor temp control or cleaning were issues.

FWIW, the engraving looks to have been done with an air assist tool, not hammer & chisel. Better closeups of the engraving would pin that down. I only mention that as the air assist/hand held tools didn't become popular/available till the mid/late 60's. Then they really exploded onto the engravers scene in the 70's with many of the people starting up at that time using them from the get=go never learning to use a hammer and chisel or a hand held push graver.
It helps 'date' work at least as 'before or after' when you can confirm the use.

Just some thoughts re: the engraving & bluing, relocated ser# ect.
Some are safety concerns/observations here that I feel should be mentioned.

"... The seller knows nothing of the history..."
There was recently a thread here on a well documented Low#;03 sporter with many years of use that suddenly let go.