Yup sounds about right, the panel to 7km sounds good. I can be swimming in the middle of the lake and get good signal on my waterproof smartphone LOL. The problem is the smart phone return will drown before it hits land LOL
Antenna gain works in both directions send+receive. So +16dBi gain in the antenna gives actually the same legal max transmit power (you have to reduce the radio dBm to be legal) , but gives a 16dB improved receive signal strength. (But as such also more interfering signal from the same direction)
Setting more TX power is only one direction, and does not extend the range if the other side is weak.
Putting a better antenna (gain) will extend the range as it works both ways, and the client is mostly the weaker radio.