What you read is completely incorrect and misleading. Power has nothing to do with overlap. The reason it appears to have have an effect is nothing more than that it may be overpowering other overlapping signals. Given enough power you will simply swamp all other signals and make yours the only one stations will receive.
There is of course a problem related to this. While the AP/base station may well be able to override overlapping signals the same will not be true of the hosts trying to communicate with it. High power only works on outgoing signals, not the inbound ones.
The relationship of power and overlap to throughput is that when a channel is "clean" the stations do not have to retransmit as frequently. The reverse of this is that where a signal is weak or being interfered with by another sources there will be a lot of retransmitting, which of course means the throughput is less than it theoretically could be.