I feel like we ought to be able to do a better job of this kind of thing with computers, without requiring the massive capital investment of a 5G cellular network to cover places 80 kilometers ("50 miles" in medieval units) from the nearest neighbor. These 11 messages from the article:

    For Armando Nahuelquir—his wife, Casilda, communicates to him that she is travelling to the campo and will arrive today.
    Inosencio Belisle, in Chubut, is advised that his brother Baldobino is sick and hospitalized in General Roca.  11 4867 5309 if he would like to communicate . . . with his niece Malvina.
    Attention Gómez Roca—we’re waiting for you in Puerto Deseado.
    For Jorge Sepúlveda in Cajón de Ginebra: Paternoster asks that he get in touch today as soon as possible, at the same number as always.
    For Jorge Crespo: Ricardo says that tomorrow, Tuesday, he will pass by his house.
    For Emmanuel Ortega, wherever he is: his sister María wishes him a very happy birthday. She cares about him very much.
    For Nicolás López, in Estancia La Elenita: his nephew Martín communicates that the four-wheel drive of the van is broken.
    For Enrique Vega, in Telsen: his mother wishes him a very happy birthday. Over here, everyone is well.
    Two years and five months since the irreparable physical loss of Fabio Ezequiel Leslie. His parents and siblings miss him and remember him with as much love as the first day. Praying for his eternal rest.
    Avelino Delgado, of Establecimiento El Calafate, in the region of El Escorial, has lost a Hereford bull. If someone sees it, please notify him on this same channel.
    For Jorge Pereyra, wherever he is: Jordan Salama, the American he met at Estancia La Adela, in Caleta Valdés, communicates that he wants to find him, if it’s possible to get in touch with the radio station. Whoever hears this, please let him know.

are 269 words, 24 words per message, but gzip to 899 bytes, averaging 82 bytes (654 bits) per message, 3.34 bits per word. (In Spanish they might be a little longer, although these English translations are already using some Spanish phrasing.) "Several minutes" of such messages "several times a day" at 150 words per minute might be 1500 words (63 messages) four times a day, some 6000 words (250 messages) per day, which would be 164000 gzipped bits at the same compression ratio as the above. Dividing that by 86400 seconds per day, we get a bit rate of 1.9 bits per second. At this bit rate, each message would take about 6 minutes to transmit if you transmitted one message at a time.

To transmit over the horizon as the AM stations do, you'd have to use similar frequencies: the mediumwave frequencies they do in fact use, or shortwave, which would permit the use of less unwieldy transmitting antennas.

So the basic setup would be: you still broadcast all the messages from a single high-powered sender, but you have some kind of analog radio receiver frontend running night and day on, say, 250 milliwatts, with a ferrite loopstick antenna like an AM radio uses. It's connected to, say, an STM32F103 (US$1.84, although here in Argentina a Blue Pill costs more like US$12 and probably comes with a clone chip with somewhat poorer analog characteristics) running at 72MHz, sampling the incoming signal at 1Msps and attempting to decode bits from it, referencing its timing to a ±50ppm quartz crystal.

The STM32F103x8/STM32F103xB dadashit tells me it uses a maximum of 50mA at 3.3 volts, so that's another 150 milliwatts. So this is a setup that probably needs about 0.4 watts, 9.6 watt hours per day, or 3000 milliamp hours per day; if you're receiving 24/7, you probably don't want to try to run it off non-rechargeable batteries the way you do with an AM radio, but a small solar panel and an 18650 rechargeable cell should be fine.

Or you could use crude time-domain multiplexing: power your receiver up, say, twice an hour for a second or two, at a time that the transmitter knows your receiver will be listening, and power back down if there's no message for you.

But it would be desirable to also transmit; it allows you to answer those messages without traveling those 80 kilometers to the post office. To get AM-radio sound quality good enough for people to understand speech, you need several thousand bits per second of channel capacity (4kHz at 10dB SNR into C = B lg(1 + S/N) gives us 13800 bits per second), and, roughly speaking, several thousand times more transmission power. https://www.lu4aa.org/wp/wp-content/uploads/2020/11/LISTADO-... tells us that LU-20 Radio Chubut, one of the radio stations participating in this practice, transmits at "25/5" kilowatts, which I think means 25 kilowatts by day, 5 kilowatts by night (that's what the same notation means in the LU-4 row, according to https://es.wikipedia.org/wiki/LU4_Radio_Nacional_Patagonia_A....)

If 2 bits per second is enough, you can probably get by with transmitting at 6000 times lower power, which is about 5 watts, or 1 watt at night. Decentralizing transmission incentivizes the use of higher frequencies, because they can use smaller antennas. At the extreme of 30MHz, a quarter-wave groundplane antenna only has to be 2.5 meters tall, and erecting even a 7.5-meter tower antenna seems eminently feasible to do on a shoestring rural sheep-herding estancia budget, since you don't have to deal with an HOA. And, if people are willing to cooperate to relay messages (a "mesh network", although the older term might be "radio relay league"), you can use lower power and get higher bandwidth.

The only part of this that would be news to radioaficionados decades ago is the radically lower price and low power of complex modulation schemes, due to cheap microcontrollers.

This is, of course, what Rhizomatica, Hermes, and MeshArg are trying to achieve, although MeshArg is using higher frequencies that are not suitable for long-distance communication unless you have a satellite or balloon or something.