Euchiton involucratus can have frilly leaves. However, it is supposed to have 3-6 leafy bracts up to 4 cm longer than the flower clusters directly under the terminal flower clusters. I would prefer to have someone look at a specimen in the hand to make a proper determination of ID. https://vicflora.rbg.vic.gov.au/flora/taxon/004eaddb-80a5-4fc5-8d99-b649eda0bab9
We have both star and jersey cudwood in our woodland and they grow quite differently and the flower heads are different. Your initial comment might be right and it is not an Euchiton at all.
Euchiton japonicus has definite globular heads at the tops of the stems, a basal rosette, and few stem leaves, which neither Vicflora not Plantnet describe as frilly. Gamochaeta americana fits better. The arrangement of the flower heads is OK, and the stem leaves are somewhat stem clasping and sometimes frilly. It usually loses its rosette by the time of flowering.