I would advise that you keep tracing the Carroll children. Their marriage certificates may well provide information about whether the father was dead or alive. In depth and detailed research will probably yield results. Also looking for newspaper obituaries for the children, can reveal information about other family members. Some family members seem to end up in Toowoomba and Brisbane, so Patrick may have died in those areas.
Some detail about ....
Thomas CARROLL born abt 1865, married 16 April 1898 where? Rockhampton? to Kate Cronin WARD
Thomas died Brisbane 1936
Children
Alice Eileen 1905
Noel John 1910; died 1984
Owen Patrick 17 Feb 1915; died 19 Feb 1915
Vincent 1917; died 1919
NOTICE
https://trove.nla.gov.au/newspaper/article/70572766?searchTerm=%22Thomas%20Carroll%22(There is a Thomas Carroll onboard the Scottish Bard 1876 age 10. That puts his birth about 1865/66 as we thought. That baptism I posted earlier may be his- just an error in mothers name.
It would be unusual to find a young lad travelling alone. He may have been accompanied by someone else. Passenger list does not help, as it is just alphabetical. I note there is a Mary Carroll age 22 (single) onboard she may or may not be related.