Being at war with a plant often means that this very plant wants to tell you something. I used to be on a war against broom and a few other plants, and then learned about their medicinal properties and practical use. After getting to know my enemy well and respecting its function in the world, I managed to let go of that 'Eliminate!!' view of things and become way less frustrated.

Which made it possible to think about more long-term solutions:

On a landscape perspective, could you offer your landscape an alternative plant? Just removing something will usually not work because landscape wants cover, but sometimes you can plant something else in the same space that will gradually crowd the privet out or better, force it back into a more manageable space. Find some use for the privet - a quick Wikipedia search finds that 'The decoction of privet leaves or bark helps to treat diarrhea, stomach ulcers, chronic bowel problems, chapped lips, sore mouths and throats, and a wash for skin problems.' (Please do some research before using a plant medicinally and start with tiny amounts. Careful the berries are mildly toxic)

For gradual removal, you can alter the composition of your soil (a gradual work) so the privet doesn't like it anymore. You can get help from allies like other plants, animals, fungi, microorganisms and add certain organic matter and/or minerals. A gradual process that will bring you very close to your landscape.

I believe quick and genocidal solutions against plants and animals who often have lived on 'our land' for longer than us is very old school. Getting to know them first and asking them politely to cede more space for our activities should probably be the way to go in the future.