Tendermint consensus is based off the solution to byzantine generals for partial synchronicity explained in this paper: http://groups.csail.mit.edu/tds/papers/Lynch/jacm88.pdf
Tendermint doesn't have a random number generator at all. You cannot have consensus on random numbers, or else they wouldn't be random. It is a fundamental problem that you are determining the next blocks winner based on numbers that can be influenced by previous winners.
If the sum is even, I win. If the sum is odd, you win.
This simple 2-person game randomly generates 1 bit. It can be extended to arbitrary numbers of people by replacing addition with XOR.
2) N people each vote either "0" or "1". They reveal votes simultaneous. The minority participants are rewarded. The median of the votes is the next random bit.
I prefer (1) over (2).
"Simultaneous" has no meaning. There are no means of instant communication, there is latency. There also is no proof of time without a consensus system.
So what is the proof that I committed my 0 or 1 value before they revealed? Well you could trust a central authority to maintain that timestamping, or you could even use Bitcoin for your timestamping. In fact, your PoS system could probably work if it piggy backed on Bitcoin completely.
If that's too abstract for you, consider that I say I'm ready to commit, everyone sends me their commits, I construct a block with all their commits and a ton of other possible commits and then once I have created a seed that allows me to win and control the network permenantly.
Thank you for the important issues you raised.