The unemployment problem is about people looking for a job and not being able to find one. If someone doesn’t even want a job though, that’s not something you can “fix” anymore.
To include people who don’t want to be employed in the labor force is counter-productive, since why would you try to find jobs for people who don’t want them?
It’s not even that these people are “lazy” like many who (wrongly) disparage the unemployed say. They just aren’t looking for jobs at the moment—stay at home parents, retired people, anybody that doesn’t need/want to work. (And by the way, plenty of homeless people receive unemployment benefits and are therefore included in the statistic.)
The unemployment rate is useful because it helps us see how many people need jobs. If you start including people who don’t need jobs, the metric just becomes less useful.
To your other point, I’m not asking you to invent a new statistic. It’s just that when people say one metric isn’t working, someone smart usually has come up with alternatives. A lot of people for example don’t like using GDP, and thus economists have come up with HDI, GNI weighted GDP, or whatever else suits their taste.
I bring this up because there are few alternatives to unemployment metrics, simply because they are not nearly as problematic as you paint them to be. Practically every competent government on Earth uses the same measures for unemployment, and specifying the labor force to only people actively looking for work is very standard; if you want to say that this nearly ubiquitous method is wrong, I think it’s reasonable to ask what needs to be fixed.