If you go to a skinnier cartridge, you will have feeding issues.
The cartridge will pop up prematurely, this may not matter on a push feed too much, but it will be an issue.
If it were me, I would spin up some scotch brite on a dowel and polish out the rust and give the throat an inspection, it may be superficial and not a concern.
You could sell it as a barreled action and keep the stock.
Cheers.
Mmm... Depends...
I picked up a LH Rem 700 LSS chambered 7mm RUM a while back. Two to three inch groups at 100 yards. Decided to rebarrel. Found a .473" LH LA SS bolt on eBay. Picked up a .30-06 CBI Remage barrel here on the 'Fire. It will feed four .30-06 cartridges right side up, upside down, vertical up, or vertical down. Repeatably. LSS RUM action, bottom metal and stock. Data point of one. YMMV...