Neither AF nor KL are flying to Australia now - both use code shares with Qantas or Singapore so no miles earned for the SIN-MEL flight.
And China Air through TPE is not an option now because Taiwan is not allowing connections.
But -
a few year ago I flew to OZ via LHR and HAN/SGN. The business class fare LHR-MEL/SYD was surprisingly cheap, about USD2300 and that included a 5 day stopover in HAN - would have been a few hundred cheaper without that stopover. Even adding in the price of US-LHR it was cheaper than a US-OZ ticket on DL.
It does mean 3 nights in the air - US-LHR, LHR-VN, VN-OZ. And you need to buy a VN visa but that is easy online and about $15 as I recall (and you need to pay with USD. VN is a good airline - you will be on either a 787 or a 350 on all legs (even the HAN-SGN leg).
It is not 7x week. I thing VN flys from LHR/CDG/FRA on different days of the week so you might want to play around.
Google flights doesn't find it as a US-OZ option. You probably have to search US-LHR and LHR-OZ separately.