I'll stick my neck out (see what I did there 😊) and suggest that in reality it makes no significant sonic difference. Just wood glue a cocktail stick or two in there, leave it to dry for some days regardless of what it says on the tube etc. Finish it flush, and rescrew. Although if planning to remove / refit neck often then inserts may be a good choice.
Note that machine screws/threads come in loose / medium / tight grades wrt thread fit. The default is medium. I suggest just giving the screwba tiny extra bit of force into the thread. Recommend hex or torx screw heads. North American people may wish to look at those "square" bit drivers.
In general they seem not to be a thing in Europe ?
And, of course, we should all be sticking to metric units now 🙂