historically we’ve been using AM at lower frequencies, and these travel further While I agree with that statement…

AM doesn’t reach further than FM

… i disagree here. Yes it kinda does, and there’s why: FM deteriorates with phase shifting introduced by phenomena such as ionospheric reflection, while AM is more resilient to it because it encodes information as amplitude variations instead of frequency (and therefore, implicit phases) variations. Also, FM needs more bandwidth than AM. Also, the overlay of two or more simultaneous AM transmissions is “more understandable” than two or more simultaneous FM transmissions laying on the same frequency. Both the three are the reasons why the modern aviation continues to use AM for comm between TWR and airplanes, as an example. Not just by historical reasons, it’s because AM is more resilient than FM.

By “reaching further”, I don’t mean the range of propagation because, as you correctly said, it has more to do about wavelength and, therefore, the carrier frequency. By “reaching further”, I actually mean the capability for the signal to be correctly demodulated and minimally understandable at the end. If a signal can propagate across hundreds of thousands of kilometers (for example, between Earth and the Moon), but it can’t be recognizable at the other point (because the phases are all messed up to the point of being unable to be demodulated), then the signal (as in the content to be transmitted/received) couldn’t really “reach further”.

Here goes an example: I live in Brazil, in the southeast. I was in Sao Paulo state (not the city) when I once managed to receive an English-spoken CB (Citizen Band, 11 meters, approx. 27MHz) transmission. Most of our neighboring countries are Spanish-speakers, the only nearest English-speaking country is Guyana (the nearest corner close to Jatapu River being 3,000 km from Sao Paulo in straight line), but I could tell by the operator accent that he was not from Guyana. The reception would be almost crystal-clear, if my receiving setup were better (I was using a RTL-SDR with a piece of long wire barely touching the outside of the antenna’s jack). While there are repeaters for CB, they’re not as common as VHF or UHF repeaters, where you can even find, for example, EchoLink repeaters, so the international transmission really made into my Brazilian home, and it was even daylight! I only could tell the signal because it was AM modulated.

When we talk about deep space communication, sure some things change, but most of the same rules apply.

These radio telescopes don’t transmit anything at all,

Back in 1974, the former Arecibo radiotelescope was used to transmit the famous Arecibo message (some sources Wikipedia and Universe Today). So, while they’re most used for reception, they can be (and they were) used for transmitting as well. It’s not a straightforward thing, it’s not simply a switch to be toggled receive-or-transmit because they involve different electronic circuitry, but the structure, the dishes and the antenna, can both transmit and receive: for reception, it just interacts with electromagnetic fields, which induces an oscillating electrical current all the way through the structure until it’s filtered (through electronic components such as variable capacitors) and amplified by a receiver circuit, while as for transmission, it conducts an oscillating electrical current and irradiates it, depending on the antenna shape and properties.

Much like a normal telescope doesn’t transmit light. It’s also a possible thing: en.wikipedia.org/…/Lunar_Laser_Ranging_experiment…