You might want to look at the 'not in labor force' numbers: https://fred.stlouisfed.org/series/LNU05026642
and also labor participation rate, but for 25-54 Yrs, so you exclude the effect of demographic changes https://fred.stlouisfed.org/series/LNS11300060
Underemployment is already reported and is distinctly different so I don't think it's fair to say that not counting someone at Burger King who has a Master's degree as unemployed is a "flaw."
The current definition makes sense because it's linked to an overt action that can be objectively determined. "Not looking for a job but theoretically would like a job" gets into all sorts of issues like "I want a job as a king if it landed on my lap...".
"Marginally attached" and "discouraged workers" are already tracked and reported in U4, U5, and U6, so this is a strange hypothetical.
It’s been creeping up for sure, but still historically low. And I’d attest that the headline rate is still the “real” rate.
Just because U3 is the measure typically quoted doesn’t mean the others don’t exist.
Source: 15 years ago I was one of the people they surveyed. Every month for a year they called me, once a month, to ask what my employment status last week was, if I was actively looking, etc. (It was all synchronized around one week a month, but I don't remember which one it was they cared about.)
The idea that the BLS lacks detailed labor market data is just internet conspiracy slop.