No Lawns

With herbicide, other than choosing the correct one, the most important part is the technique in application, the amount irrigated into the wound (without spillage), and how quickly it is done. I've seen plenty of people treat things and then step up to worse herbicides because they didn't do it properly in the first place. Ideally, you would learn how to ringbark (Americans use girdle but there are 3 types of American girdling under that definition that all relate to trees and no way to tell them apart), which would have removed all need for chemicals. It's too late now. The chemical free way from this point on is excavation or light/sugar starvation.

You've been suggested a drill and that is how most standing trees are done these days. Drilling into the roots and/or root plate is reasonably standard. Again, it's technique not just drilling holes deep into the tissue (waste of chem). I will edit in a pic that is somewhat related, you are conducting different works now but the pic would have been how it was done to start with. A chainsaw is fine, just a good irrigation of the correct chemical ASAP should have worked. Shallow drill holes with a 10mm-ish metal drill, staggered through roots that are shooting or alive, filling up the hole twice with chem (fill hole within a second or 2, wait, fill it up again before 15 seconds expires).

Here is our fact sheet on control methods:

https://www.daf.qld.gov.au/__data/assets/pdf_file/0020/77420/small-leaf-privet.pdf

Ignore the text, that's just me not cropping it properly: