It's a perfectly good statistic for economists, because they know those limitations. There are lots of other measures for wages, labor participation, etc, and 'unemployment' does capture useful info about the ease of finding a job after you leave a previous one.
It's just abused in the public sphere as a whole-story statistic.
It is perfectly good statistic for POLITICIANS because 4.4% uneployment rate does sound much better that 22.1% reported by ShadowStats. Former is typical for rich (e.g. G-7) countries, latter would make US look like 3rd world country in that regard.
You might be right that it does capture info about how hard it is to find a new job as the software developer, or a lawyer, but tells almost nothing about blue collar jobs.
At best it allows you to compare snapshots in identical situations over very short time periods assuming nothing significant is going on.