The worst players I've seen in this game have all been on PBE. Leaving in incredibly frequent. Why is this acceptable? Why is there no punishment?

they aren't, Riot has closed PBE signups specifically to crack down on this issue.

In this last game, I had an MF go mid with me(because she typed "mid" .2 seconds faster than I could say "mid please" after I already picked a champion), and then call me trash noob **** all game--literally all game--because my ignite's last tick got first blood. Then, the enemy Katarina was trash talking her team in all chat(while MF was trash talking me in all chat), our Skarner started raging and chat was filled with NOOB NOOB NOOB NOOBS LOL NOOBS

In the game I just played, I got called **** and worst Riven NA because our support blitz kept hooking their fed Xin Zhao while I tried to even out a 2v1 top. He then went AFK for several minutes before deciding that intentionally feeding would be a better way to spend his time.

These players are allowed to do whatever they want because there is no punishment for their actions on the PBE. That is unacceptable.

Waaaahhh

This happens because PBE is just that, a Public Beta Environment. It's not designed to have proper matches or replace the normal regions for any reason, it's there for people who either want to test out potential changes or contribute their opinion to a future bug fix or change. For those (like yourself) who want to play it as a normal region because of all the free RP/IP then your going to have to put up with ****