I would very carefully weigh your plan.
Its your rifle,you can do things your way.I'm sure it has been done before,I won't tell you it can't be done.
First,you need to evauate your serial number range.Some Springfields are "low number" rifles,and even shooting them with 30-06 is questionable to unsafe.
While I am not saying the 1903 cannot handle the pressures,SAAMI maxpressure for a 300 win is 64,000 psi vs 62,000 for the 30-06.The larger diameter of the case head translates to a little more bolt thrust.The cone breech leaves more brass unsupported than some other designs,which can be important when case stretch weakens brass .
Is it in original issue trim?If so,the surplus import laws say there won't be any more coming back to the US,there are no more Springfields.If you alter it,you will greatly decrease the value.What may well be a $750 rifle original(conservative estimate) will become a $250 rifle,if you spend at least another$400 or so on it.
You may well have the feed rail issues that concern you,ignoring that,add the price of:
bolt handle job for scope clearance,a new safety,as the original won't work with a scope.Drill and tap for scope mounts.Do you have a chamber reamer and headspace gages?
Unless you go Adams and Bennet or equivalent,barrels start at about $225 and go up to about $350 if you don't get too fancy.
You might want to go to the Winchester web page,I guess they are back up and running,and see what a new M-70 will cost you in 300 Win mag.
I think if you do the figuring,for the same money,you can keep your original 1903 in 30-06,and buy a new M-70.
I have sporterized a number of milsurp actions and built rifles.It is something I have enjoyed,and learned from.I don't tear down rifles,I buy a bare action when I find one.Today,if I had a good 1903 action,I'd get a Kreiger/Criterion original contour 30-06 barrel,one of the new replacement stocks,or I might find my own piece of good american old dense walnut and have an original made on a duplicator.Then I'd scrounge the rest of the furniture to make a mil trim 1903.In 30-06.