Mike
Congratulations on doing it all by hand: No easy job, but very traditional. One day I must learn how to do that video thing, but thank you for linking. Of course a lot depends on how much of this type of thing you plan to do, but once started it is a little bit drug like.
Just a couple of easy things: You probably need a wider screwdriver for undoing the saw screws as the brass tends to damage rather easily if it is the slightest bit tight. Most commercial drivers are too narrow. You can fashion something fairly easily out of an old piece of flat file or one of those rubbish quality chisels can be reshaped and used. I have two different sizes as the saw screws on hand saws are larger. It helps to have the blade nearly the width of the saw screw.
The female screws are usually easy to get off, but the male half as you found out can be more difficult. I just use an appropriate size nail the reverse way around and tap gently from the reverse side until the screw releases. In your case you were replacing the handle so damage was not an issue. I have glued a few different size nails into handles but it is not necessary either to have a handle or different sizes. I only ever use the one size! The handles prevents the nail getting lost. A suitable sized pin punch would also do the trick.
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It is always going to be tricky drilling super clean holes for the saw screws and the medallion with hand powered tools. Forstner bits are excellent, but I also use brad point bits and spade bits. Spade bits are quite easy to modify if the size is just too large. I have at least a couple modified in this way. It is advisable to identify that they are no longer the size stated in the metal. I use a paint marker.
Of course you are going to have to sell your soul to the devil and use an electric drill (a drill press is much better again).
I don't think you mentioned the timber, which looks to be Silky Oak, and that is easy to work if a little bit on the cheesy side. If you had used a harder species you might have struggled with the sanding operations. That would be where good rasps come into their own, but they are expensive. Also I cut off the bulk of the waste for the rounding with either a turning bow saw or a coping saw.
I also like to remove the back but that is another step and an added complication.
Regards
Paul