isotropic antenna recieved power
If the antenna was amplifying the signal according to the antenna gain then yes it could receive more power, but common sense tells you that it doesn't seem right. The antenna itself cannot amplify a signal, all it can do it try to direct most of the energy in a specific direction. An isotropic antenna is a theoritical antenna that can radiate power equally in all directions. The gain of an antenna is the maximum amount of power it can radiate in specific direction compared to the power an isotropic antenna can radiate in any given direction (since they're all equal).
Your calculations forgot to include the free space loss which is considerably more than the gain of the antenna and is proportional to r^2, r being the distance between both antennas.
Look up or google Friis Transmission equation for more info on the subject.
I hope this helps.