You're mixing up constitutional obligations with public expectations.
Yes, Catherine is a publicly funded figure in the sense that she is supported as part of the Royal Household. But she has no constitutional role, no legal duties, and no salary in her own right. William is the constitutional office-holder as heir to the throne; Catherine is his spouse.
A "working royal" isn't a legal office with mandatory hours or a contract. It simply means someone undertakes official engagements on behalf of the monarchy. Catherine chooses to do that.
Saying her role is "useless" ignores the work she actually does. Her early years campaign led to the Business Taskforce for Early Childhood, businesses invested millions and created hundreds of apprenticeships, she brought together companies to donate over 10,000 essential items to baby banks, and the Royal Foundation invests millions every year in charitable programmes.
As for William "barely doing anything," that's simply an opinion, not a fact. His work includes Earthshot, Homewards, Duchy of Cornwall responsibilities, investitures, overseas tours, military appointments, and official engagements. You may prefer the old model of hundreds of ribbon-cutting visits, but they've made it clear they want to focus on fewer engagements with longer-term impact. You can disagree with that approach, but it's not the same as doing nothing.