That's not an assumption, it's a simple fact.
http://lmgtfy.com/?q=teacher+salary
Teacher wages are not even close to poverty, and they get lots of non-wage compensation which costs real money (not to mention they only work 9 months/year).
Politically connected insiders are less relevant than those who actually have the majority of the money.
On the contrary, the existence of politically connected insiders mooching off the system is evidence that we can provide more assistance to the poor without raising taxes.
http://lmgtfy.com/?q=income+inequality+in+america
That puts them well below the mark where they'd need to be to be to place in the top X% (10-30, depending on what study you use) that own 80% of American wealth. If you believe that shuffling around the remaining 20% will work, more power to you. But I'd have to disagree.
I will, however, agree that politically connected insiders mooching off the system is DEFINITELY an issue in the US. I would just point to Corporations and the Rich in place of Teachers on your list.
The fact that teachers may receive average pay for below average work does not put them at the "brink of poverty".
As for finding the insiders, follow the money. Teachers and old people are some of the biggest recipients of government money.
There are ALWAYS more assumptions available to be challenged. Plenty in your post for instance. Turtles (assumptions) all the way down.
For instance, if the goal is to reduce poverty, you assume that it is mathematically possibly to take money from teachers. But if teachers are already on the brink of poverty, that won't work to reduce poverty. You're creating new poverty cases as you solve old ones.
Politically connected insiders are less relevant than those who actually have the majority of the money. It's not teachers, and it's not unions. It's software entrepreneurs, MBAs, and oligopolists. They have the far majority.