There are some problems with your plan that others have not mentioned. Although the bolt face is the same as many other hot rod 22s, the case base diameter is not.
The 220 swift is a rimmed cartridge that headspaces off the rim whereas most other wildcats and standard case bases headspace off the shoulder.
That said, the headspace issue can be handled by a decent gunsmith. As an amateur Smith myself, I wouldn't hesitate to tackle that part of the job. What would stop me cold in the water is the potential feed problems associated with the smaller diameter 220 case. Unless you want to convert the rifle into a single shot, there is no point in changing the headspace method if the new standard diameter case won't feed properly.
DO NOT LET ANYONE tell you that all Remington 700s Short Actions have the same feed rail dimensions. Some do, and some do not. This issue is not well documented. I see it regularly. I would fully expect that an action designed to feed the 220 swift would have narrower feed rail spacing and perhaps even a different axial rail ramp angle. I DO NOT know that for a fact for the 220 swift, but I would fully expect it. I certainly wouldn't chamber a new barrel in a different caliber and change the head spacing without carefully checking that issue.
So, assuming that works, my own preference for rechambering (if that's what you want to do) would be a 22 BR. The 6 & 22 BRs were developed by Remington for Benchrest competition (hence the designation "BR") around the same time that the 6 & 22 PPCs were developed. Although many forum members will take issue with calling the BRs & PPCs "inherently accurate", they would be hard pressed to argue against their overall success in benchrest competition or the relative ease of developing highly accurate loads for them. To be fair, the 22BR is not as fast as the 220 Swift, but it's close enough that varmints will never know the difference. More importantly, the short fat BR case is much kinder to a barrel throat than the swift could ever be.
But that's only if you insist on a new caliber. My personal advice to you is to skip over all those issues and either install a new barrel in 220 Swift or simply set your existing barrel back a half inch or so or whatever is needed to clean up the throat erosion. Better yet, set the barrel back, shoot it till the throat erodes again, and then install a new barrel when needed. There are lots of other advantages too - reloading equipment, cases, and the exotic thrill of the Swift itself!