They physically couldn't.
Think about all the changes in goaltender protection technology over the last 50 years - how much pads have changed, protectors, waffles, gloves, masks, etc.
It wasn't until the early 80s that goalies really had the physical tools to last for as long as they do now. It's why players like Sawchuk and Esposito are so revered - they just seemed to be made of steel.
I'm not sure that's entirely accurate. The top-20 in career goalie GP includes Sawchuk (1950), Hall (1952), Esposito (1968), Worsley (1953), Plante (1953), Lumley (1944), Vachon (1967), and Meloche (1971). That's 8 out of the 20 biggest-career goalies who started their careers between 1950 and 1971.
I think the real explanation is simply that the league expanded and allowed goalies on the downside of their careers to continue in the NHL. If you take a random season, 1951, and look at the starting goalies in the league at the time:
Jack Gelineau (BOS) - Age 26
Harry Lumley (CHI) - Age 24
Terry Sawchuk (DET) - Age 21
Gerry McNeil (MTL) - Age 24
Chuck Rayner (NYR) - Age 30
Al Rollins (TOR) - Age 24
Turk Broda (TOR) - Age 36
That's a pretty ordinary spread of ages for the Original Six era, basically a bunch of guys in their physical primes plus the veteran Rayner and the physical fluke Broda. Now consider that the following season Gelineau lost his spot to "Sugar" Jim Henry and spent the next 4 seasons in the minors; Rayner managed one more season as starter before he was bumped to a backup role and then to the minors by a young Gump Worsley; McNeil was demoted 3 years later when Jacques Plante took his job; and of course Broda retired because it was time.
Then look a few years later at 1955:
John Henderson (BOS) - Age 21
Jim Henry (BOS) - Age 34
Al Rollins (CHI) - Age 28
Terry Sawchuk (DET) - Age 25
Jacques Plante (MTL) - Age 26
Gump Worsley (NYR) - Age 25
Harry Lumley (TOR) - Age 28
Basically the same age range, right? Nothing much has changed in terms of entry to the NHL, as guys in their early-mid 20s dominate the starting positions and older guys are lucky to get a platoon job.
So what's different with this group? Henry, of course, retired and Henderson wasn't an NHL quality goalie so he was back in the minors quickly. Rollins' career was cut short when he was exiled to the minors after a contract dispute (another aspect that would change dramatically over time). Lumley played one more season as starter and then a few years as a backup before retiring at a then-ordinary 33. So the older guys and the sub-NHL'ers were weeded out quickly.
But Sawchuk, Plante and Worsley were in the "sweet spot" with regard to career timing. Sawchuk's last season as a full-time starter was at age 31, which was normal... then he played a few more years as a platooner and then a backup, which was also normal... then in 1968 the league expanded and he signed with the Kings, and then the Wings, and then the Rangers to finish his career. Meanwhile Plante made it to an impressive age 35 as a starter, and retired... only to come back in 1969 in a platoon role with the Blues for 2 years, then 2 1/2 more years in Toronto, then a cameo in Boston, and even a stint in the WHA. He retired for good at age 46. Worsley's career looked finished at age 33, though he held on to take a one-off starting job in Montreal a couple of years later, which eventually led to him playing
five more years in Minnesota after expansion. In all three of those cases, the goalies ended up with careers which were unthinkably long by the previous generation's standards -- not only because they were great goalies, but also because the league doubled in size just as they would have been retiring.
Look at the other guys on the list... Rogie Vachon started in the O6 and by the time he was a veteran there were 20 NHL teams and 14 WHA teams offering jobs. When he stopped being useful to Montreal, he took a job with a terrible LA team instead. After he broke into the NHL, he never set foot in the minors again. Gilles Meloche, same story. He team-hopped but never left the NHL.
So by the time you get to the 80s, being sent to the minors wasn't even a consideration for a decent goalie. It was just assumed that if he was NHL-quality to begin with, a goalie could always find a job with a basement team and extend his career by a few more years.
By the turn of the century you had almost 100 goalies cycling through the league every year, so losing a roster spot was out of the question for even a Salo or Shields level goalie. That's why you have stuff like Sean Burke and Nikolai Khabibulin being near the top of the all time GP list, and close to leading some franchises.
Sorry, didn't mean to write so much about this
The more I looked into it, the more the granular details started to seem pertinent.