The reason you use a metal drill bit and not a wood drill as a wood drill bit pulls into the wood, the metal drill allows you to dictate how far you go with a in/out motion. Doing roots, you usually only have access to the top and a little bit of the sides so the cambium is quite thin, you can make small “cups” with the drill on the top and sides for “injecting” the chemical. Use a nozzle or spray bottle, a sauce bottle might work, a dropper etc. Don’t spill it out of the cup you’ve made or it’s wasted.

The secret to killing Privet is Metsulfuron but I wouldn’t use that where any food production was happening. Privet hates Metz.