
I appreciate you contacting me. I also appreciate the difficulty in identifying the road name accurately.

To the best of my knowledge, the NSW Address Location Service is the most accurate source as it evolved from confusion over street names and the inability of emergency vehicles to locate some addresses in emergencies. The NSW Govt established this service to ensure all emergency responses had accurate addresses. I recall checking this some time ago and could find only one property using this road as an address and it still showed as valid this morning when I checked using the URL : 
https://maps.six.nsw.gov.au/services/public/Address_Location?houseNumber=2031&roadName=Burke  Wills&roadType=Road&suburb=Balranald&postCode=2715&projection=EPSG%3A4326

The address service gives no confirmation for a property on "Burke and WIlls Road", "Burke & Wills Road" or "Tin Tin Bidura Road".  It is unusual for road names to include "and" or "&" ( I can't think of any other one at the moment)  and perhaps the NSW Address Location Service database cannot cope with "and" or "&" in road names. 

Local governments are the authorities in regard to road names in NSW. In this case, Balranald Shire Council still identifies this road as Tin Tin Bidura Road (I checked the Council's Register of Roads on its website this morning)

It might be best for OSM to show all three names, one as primary name and other two as alternatives so that a search for any of the three names will take a person to this road. I'm not sure how to show two alternate addresses in OSM
perhaps 
name=Burke & Wills Road
source:name=signposts at each end of road
alt_name=Tin Tin Bidura Road
source:alt_name=Balranald Shire Council
old_name=Burke Wills Road
source:old_name=NSW Address Location Service

I was pondering sending an email to Balranald Shire Council but, based on past experience, I expect they will respond with the official name that is in their Register of Roads i.e. Tin Tin Bidura Road so it would not clarify the issue for us. (Bidura is a property near the northern end or the road while Tin Tin is a property in the vicinity of Pitarpunga Lake)

My preference is to include all three names in some way but I am happy for you to make whatever changes you think appropriate.  I have seen some of your work in OSM and I know you are a conscientious mapper and I trust your judgement. You do what you think is correct.