Short answer
In 1972, there was no automatic position reporting, this had to be done by voice when the aircraft was not in radar contact. Distress transmitters, which range can reach 150 km in favorable circumstances, were already used by military at some places, and were being generalized to commercial aircraft. However distress signals were not monitored by satellites like they are today, but from ground stations (including by ham radio-operators who since a long time play a significant role in disaster communications and localization), from airborne aircraft and from ships.
If the search team couldn't find the aircraft, it can be because no distress beacon was on the aircraft, because they searched at the wrong place, or because for some reason it was not active. Beacons of that era could fail simply because their antenna was destroyed, or shielded by metal debris. Automatic beacons could also fail to activate, and the team may be unable to trigger it manually because they cannot recover it.
You mention GPS, be aware GPS is of no help in position reporting. Like other positioning systems, GPS can be used by the crew to determine their position. They still have to report it to ground control by some means.
Details
Position determination vs. reporting
It is important not to not mix two different things: Position determination and position reporting. Today you may read everything is "GPS tracked", but this is inaccurate and misleading. While many systems track us in real time, like the cellular phone operators, this is not the case of GPS, because a GPS receiver is a just a receiver, it doesn't send any information nor interact with satellites.
In the case of aircraft, ground control using a radar is able to determine the aircraft location and can share it with the crew. Radars have a limited range, don't work well in mountainous area, and are not implemented in oceans. When no radar can be used, the aircraft crew themselves determine the position. Then this position must be reported to ground control to make them aware.
Position determination and reporting by aircraft
Position determination. Today includes GNSS (GPS, Galileo, Glonass, etc). At the time this happened GNSS didn't exist, but other techniques were used, most still used today: Inertial system, Loran, VOR, TACAN, etc.
Position reporting. At the time, when out of radar coverage, reporting was done by the crew by voice communication, at predetermined reporting points along the route. The need to report the position is part of regulation, e.g. for FAA in FAR 71.5:
§ 71.5 Reporting points. The reporting points listed in subpart H of FAA Order JO 7400.11F (incorporated by reference, see § 71.1) consist of geographic locations at which the position of an aircraft must be reported in accordance with part 91 of this chapter.
Today there are multiple techniques to automatically report the position, like CPDLC, the most used is ADS-B. ADS-B is coupled with a GNSS receiver. The GPS position is broadcast blindly by the ADS-B transmitter and can be received both by control for tracking and by other aircraft for collision avoidance.

Current integration of SSR (radar) and ADS-B/TIS-B, source
As ADS-B is not encrypted, actually anyone can receive the aircraft position, and this is what ADS-B tracking sites use first.
There must be ADS-B ground stations closer than radio horizon. So ADS-B suffers the same range limit than radar. But this is going to evolve quickly, in the next 3 to 5 years ADS-B will be received by satellites, and satellites will relay the reportings to ground control. See details below.
When the aircraft position cannot be tracked continuously from the ground, another possibility can be used when a crash occurs: A distress radio signal can be transmitted on a special frequency.
Distress transmitters
The distress transmitters have generally an automatic switch detecting the large deceleration associated with a crash. This radio signal doesn't need to transmit the position which can be determined by several techniques, from triangulation on the ground to Doppler evaluation from satellites.
Distress transmitters are known under different appellations, the one used by aviation is emergency locator transmitter or ELT.
ELT
At the time this accident happened, ELT were used by US military aircraft, perhaps not by Uruguayan military aircraft. In the early 70s the system was also required on commercial aircraft.
At this time ELT transmitted on the distress frequency of 121.5 MHz could be received by stations closer than the radio horizon. That could be ground stations or airborne aircraft, mainly those crossing the oceans. In non populated areas, a beacon triggered within a radius of say 150 km from a flying aircraft could still be detected and the information relayed to search and rescue organizations.

121.5/243 MHz ELT, source
To locate the ELT, search teams used triangulation techniques similar to direction finders used both by ground control and aircraft.
ELT have evolved to a detection from satellites which permanently monitor distress signals at any place. Triangulation techniques have been replaced by Doppler sensing based on the satellite velocity relatively to the searched beacon.
Such ELT transmit an identifier at time intervals on 406 MHz, a frequency constantly monitored by Cospas-Sarsat, a system made of two constellations of satellites, one in geostationary orbit at high altitude, and one at low altitude but constantly moving.
Most 406 MHz ELT are coupled with a GPS receiver to autonomously determine their position. The GPS-based position is transmitted on the 406 MHz signal, double-checked by Doppler sensing, and reported to the search and rescue organization.

ELT using GPS and Cospas-Sarsat, source
They still send a continuous signal on the old 121.5 MHz frequency in order to allow the ground search team to use short-range direction finders to reach the ELT.
ELT limitation
While ELT may replace continuous tracking for the purpose of search and rescue, this is still unsatisfactory, as both MH370 and AF447 accidents have shown:
- In some crashes, ELT were not triggered.
- ELT cannot work underwater
- ELT may be destroyed in the crash or their signal shielded by debris.
There are some ELT which can be released before ground contact, especially on helicopters. However technology allows for better solutions, but this assume a significant change in existing air traffic control organization, switching from ground-based control to space-based control.
ATC switching to space-based ATC
Current aviation traffic control relies mostly on ground stations. When the aircraft flies over oceans the communication is broken (or at least unreliable).
To solve this problem with modern technologies, plans are being implemented, both in Europe and North America, to use satellites to track aircraft and these plans include a generalized use of ADS-B transmission by aircraft.

Thales / Aireon solution for space ATC, source
In anticipation of this deployment some operators are already using satellites to communicate their position at regular interval. They don't sent it to ATC since ATC is not ready to process it, but to their own operation control center. This was a lesson learnt from MH370.
MH370 had all the technology required to determine and report its position:
- GPS and inertial navigation system for position determination.
- ACARS to report its position.
- a satcom link to transmit over water areas (the satcom transmitter position was determined by Doppler sensing, this led to searching water areas).
However the ACARS was configured to transmit only to ACARS land stations using VHF links, and was prevented to use the satcom link, for economical reasons as the flight was supposed to happen mostly over land.