So if i’m despondent and give up searching for a job then doesn’t U-3 unemployment go down and therefore wouldn’t using a definition that included the despondent not be linearly dependent?

Though it seems hypothetically possible for what you're saying to happen, the data shows there's a linear relationship between many different definitions of unemployment throughout the years in the US.

https://www.bls.gov/opub/ted/2018/u-3-unemployment-rate-was-...

Yes, if your job is to track that analyze it and report on it.

But U3 is fine for year-to-year readings of employment temperature for lay people like most of us.

I want to add that "actual employment rate" or labor market participation rate is also a grossly misleading figure. For example America had it's highest labor market participation rate in 2000 - during the great recession. Also, Indonesia has a much higher participation rate than America and most Western countries, even though the job market there is likely not as good as Europe or America. Russia is above Germany despite the fact that any reasonable person would choose to job hunt in Germany over Russia. Unemployment does actually seem to be the better metric