I agree with qbilder, this is the best method for removal. It leaves the rings intact and simplifies the installation on the new shaft.
There is a concern as to the pin size that you'll be going with though. This will have a bearing on the diameter bored in the ring-set.
If you're choosing a pin that requires an insert, the insert will have an outside diameter of .437" (7/16"), Uni-Loc .445". With a 1/2" tenon to support the ring-set your wood wall thickness btwn. the ring-set and the insert would only be .027"/.032". Boring the ring-set to .620" would increase the wood wall thickness surrounding the insert to .0915" avg.
A 3/8" pin combined with a 1/2" ring tenon yields a wood wall thickness of .062". This may not be a big deal because most 'big' pins are usually longer than 1" exposed.
Another thought, how tall is the ring stack? How close does it get to the bottom of the installed insert? The closer it gets, the more the integrity of the shaft is relying on the unsupported wood wall thickness.
These are just some variables that may or may not come into play and are only offered for your consideration.