Others here have the $9895 - $11,169 (EuroOptic) for an AXSR or the $4250 - $5250 for an AT-X; not including tax. Not everyone has that kind of disposable income. The others have correctly stated if your goal is a real AI rifle, then buy a real AI rifle. If the goal is something to build up to as you can afford in the end, unfortunately, AI does not support such a piecemeal approach and wants you to go all in, money-wise (eBay does have AI action-specific chassis that are hard to sell because people tend to go for the Remington clone AI chassis options instead).
The factory AI's have a flat bottom square receiver action with a specific interface that only works with an AI action-specific chassis they are mated to; they will not work with other actions. If your goal is a non-AI action, like a Remington footprint or a non-Remington footprint like a Surgeon 1581XL, then yes, your approach is 'feasible' (AXSR go for about $1500-$1700 used on eBay or the PX, if you can find, with retail from $2000-$2760). On the low end with an AXSR chassis and a Remington 700 barreled action swapped in, you are at about $2700-$3300, all in, not counting scope. If you get any aftermarket action like a Defiance, Terminus, BigHorn, Curtis, ARC, Origin, etc., you are, like the others said, up at the price of an AT-X even with pre-fit barrels that need no gunsmith.
So given this info and what the others said, if you are ok with a base Remington 700 in an AXSR chassis, you will save a grand on the rifle set-up alone. For one more grand, you get an interchangeable barrel system on the AT-X (seen for $3250 as a base auction price on GB by MileHighShooting over the holidays, but in red color, though), but I presume that is not the 'look' you are after. If you are on a budget and are going for the 'look,' then, yes, the AXSR chassis option is feasible given these considerations. Cheers!
