A slight variant on your fine answer...
A reference is Ramo et al, Fields and Waves in Communication Electronics, chapter 12.
First, reciprocity: $Z_{21}=Z_{12}$ tells you that (assuming a conjugate-matched load):
$$ g_{dt} A_{er} = g_{dr} A_{et}$$
For both transmitting (subscript t) and receiving (r) antennas, $g_d$ is the antenna directional gain.
$A_{er}$ is the effective area of the receiving antenna, defined as the ratio of useful power removed from the receiving antenna $W_r$ to average power density $P_{av}$ in the incoming radiation.
Thus the ratio $g_d/A_e$ is the same for both transmitting and receiving antennas.
For large aperture antennas, it can be shown that the maximum possible gain satisfies:
$$ \frac{(g_d)_{max}}{A_e} = \frac{4 \pi}{\lambda^2} $$
For other geometries, $A_e$ is defined to give the same result. For example, for a Hertzian dipole, with a maximum directivity of 1.5:
$$ (A_e)_{max} = \frac{\lambda^2}{4 \pi} (g_d)_{max} = \frac{3}{8 \pi} \lambda^2 $$
Anyway, for the problem at hand, as you deduced, the useful power removed from the receiving antenna is:
$$ W_r = P_{av} A_{er} \text{, with the power density } P_{av} = \frac{E_b^2}{2 Z_o} , Z_o=377 \text{ ohms} $$
(Here, electric field and voltage are sinusoids measured as peak values.)
With a conjugate-matched load with real part $R_L$, equating load power dissipated with power delivered gives for the receiving antenna's Thevenin equivalent source voltage $V_a$:
$$\frac{(V_a/2)^2}{2 R_L} = \frac{E_b^2}{2 Z_o} A_{er} $$
$$ V_a = 2 \sqrt{A_{er}} \sqrt{\frac{R_L}{Z_o}} \, E_b $$
Substituting for $A_{er}$ from the reciprocity relation, the maximum voltage $V_{a,max}$ is:
$$ V_{a,max} = \sqrt{\frac{(g_{dr})_{max}}{\pi }} \sqrt{\frac{R_L}{Z_o}} \,\, \lambda E_b $$
I'm cautious about the $\cos \psi$ factor because beam patterns differ for different antennas.
Best Answer
The electromagnetic field radiated by the transmitter (and every other transmitter and noise source) causes current to flow in the antenna. This is filtered and amplified by the radio. The radio needs to weed out the signal it's looking for from all the rest of the junk coming in on the antenna.
Normal antennas aren't as selective as you are expecting. A resonant antenna can be somewhat selective, but it requires special construction. A ham antenna called a magnetic loop is about the most selective real antenna I know of. They are employed at tens of megahertz.
The degree of selectivity is referred to as $Q$.