To answer your last question first - no, on the whole it doesn't. But this isn't a present problem. This hollowing out of the middle has it's genesis around the Carter administration (not saying Carter was at fault, just the timeline). This is nothing new and is a chronic issue. To be made worse with automation (idle hands problem).
This is why I have always liked the U6 measure of employment rather than U3. U6 does look at underemployment and other items and is the most conservative measure. Right now it's at 7% or so, which is actually quite good. And, yes, you are correct that jobs are counted as jobs. U3 and U6 employment numbers don't dig into distributions of jobs, etc. (Employment is actually a really deep, hairy subject). If you want some good discussion of employment numbers I find
Mish Shedlock to be really good. Just note he's a permabear.
Having said all that about the long term issues of job quality my post you quoted is more about more present comparisons, which do look quite good.
DJT can be blamed for a lot of things, but the long term structural decay of middle class employment isn't one of them.
