Just because he didn't use any form of degreaser doesn't mean that you can't or shouldn't. How effective a degreaser will be in each case will depend a lot on how often the stock was given a "feeding" coat of oil and if the oil used was the proper boiled linseed, tung or other oil that hardens over time in contact with air.
But the real reason for degreasing the wood is the oil used on the metal parts. That oil does not harden, But it does happily soak into the wood. Cleaning all traces of the metal oil away is what the degreaser is best at doing and why it should be done.
Some brass through screws isn't a bad idea either if you want such a fix. But I suspect most would LIKE to get away with just a glue repair to retain the original look as much as practical. Keep in mind that the through screw style repair is pretty much a belt and suspenders approach and as such would be highly respected on a battle field weapon where you would not want to only rely on some glue holding things together in wood which is highly suspect for it's ability to glue given oil, water and other issues found in a war zone. But with us if the joint SHOULD let go at the range it's not a big deal to set the gun aside and for the second time repairing the stock to go with the screw option.