Yes and no. I emphatically agree that the way the Path Loss Equation (Friis) is taught is misleading. I much prefer the way you interpret it, with the transmit antenna represented with gain and the receiving antenna having only an effective receive area. It's much more intuitive because I can visualize a spherical shell of power radiating outward.
That said, a receive antenna does absolutely have "gain", which is evident by the antenna receiving a stronger or weaker signal depending on its orientation with respect to the transmit antenna. The key is this: for an arbitrary antenna, the (transmit, if you like) gain has a one-to-one relationship to the "effective receive area" at a given frequency, so talking about area and gain are equivalent, if not intuitive. We usually assume for point-to-point links that the antennas are oriented at each other, and in such cases (for good aperture antennas), you are absolutely right that the physical area and effective area are approximately equal. For ideal wire antennas, however, the physical area of the antenna is 0, but the effective area is nonzero (because of magic).
Now, I disagree that the path loss equation violates conservation of energy. The link to the effective area and gain depends on the wavelength. When I increase the frequency of operation but I keep the gain of the antennas constant, the areas decrease, so my receive antenna is physically smaller and the power goes down. Not breaking physics. A lot of people will say "path loss gets worse as you go up in frequency", and this is extremely misleading if not "scientifically illiterate" as you pointed out. Sure, there are molecular absorption bands from oxygen/water that literally dissipate power in the atmosphere, but generally speaking, the path loss didn't get worse, your receive antenna just got smaller.
Now wait a minute, what if I just made my receive antenna larger? Well, you can do that! The problem is that because gain and area are linked, efficiently receiving power in a given LARGE area (with respect to the wavelength) implies high gain. High gain implies a very narrow beam (more like a laser pointer than a normal dipole spilling energy everywhere). So it becomes really important that I "point" my receive antenna perfectly at the transmitter. Satellite dishes are really big, and they absolutely have to be pointed accurately at the satellite.