I am a believer in wooden match stick ends. Cram 1 into hole and break off flush. I don't know how many times I have done it over the past 40 years.
-bernie
I agree with Bernie I've always used tooth picks with a little wood glue. Its works great
When I started woodworking, I didn't know squat. I have progressed in 30 years - now I do know squat.
This hole is a bit precarious. Directly behind the hole is where the bolt sits. So I cannot push whatever will fill the hole all the way down or it would interfere with the locking bolt. It's likely the locking bolt can be removed and then the finger will be used to feel for the depth of the material inserted to close the hole. Lots of effort for such a small task.
Another item that I like to use is bamboo skewers. They also work well
I have the Lee Valley item shown above and it is great if you have enough thickness to use it on. From your shallow requirement my first choice would be to drill a 1/4" flat bottomed hole with a Forstner, glue in a short piece of 1/4" oak dowel, drill and screw. If the holes are not too over-large, whittle a plug of hardwood scrap, use a toothpick, a kabob skewer, etc. and glue in place. Saw off flush, drill and screw.
Durham's is a great solution for many things. It is a poor wood filler due to the reason you state; it is harder than the surrounding material and therefor is difficult to sand flush. It doe not shrink appreciably though so you can putty-knife it flush while soft, then drill and screw into it.
"A hen is only an egg's way of making another egg".
– Samuel Butler