Did ice-hockey players actually not wear helmets until the 1970s-1980s?

NHL players started wearing helmets for the most part after a player, Bill Masterson, died from a check that caused him to hit his head on the ice at speed. That occurred in 1968. The mandate was about a decade later:

[I]t took another ten years before the then-President of the NHL, John Ziegler, made protective helmets mandatory for all new incoming players.

So - yes, it was really in the late 1970s before all players were wearing helmets.

From an SI piece:

The NHL made helmets mandatory four decades ago. Any player who entered the league after June 1, 1979 had to wear a helmet, but any player who signed his first pro contract before then could opt out if they signed a waiver. During the 1978-79 season, about 30 percent of NHLers didn’t wear a helmet. Ten years later, though, and you could count on both hands how many helmetless players were left in the league.

Craig McTavish was the last player to not wear a helmet, who left the league after the 1996-1997 season!

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Unrelated note: please do wear a bike helmet! Nearly 1000 people die annually from bicycle injuries. Most of those are from head injuries - and of those who died, 97% were not wearing a helmet!

For the statistically inclined: a study in 2006 showed a 51% reduction in Traumatic Brain Injuries and a 44% reduction in mortality in helmeted riders.

Yes, and it's not because they were especially stupid. People even a couple generations ago were dramatically less risk-averse than today. They accepted a certain risk of injury as part of playing professional sports, and they didn't consider the risk of a head injury as being particularly high (until, as Joe mentions, one incident caused a shift in perception). Taking special precaution against it was indeed seen as a bit silly. As you say in the question, when major consequences are rare enough, most people are happy to believe that it will never happen to them.

As quoted in the SI article Joe linked, players felt that helmets reduced their field of vision, and so their situational awareness. What's more, the "sissy" argument actually holds a bit of water in competitive sports. If helmets are optional, if you're wearing a helmet and most players aren't, that signals to the competition that you're more concerned for the safety of your head than most people are... which will probably inspire them to threaten your head (as much as they can get away with) in order to mess with you, and they'll have less reservation about doing so because you have protection (again, from the SI article, Greg Smyth felt that "players might have been a bit more tentative" about attacking him during the games when he went helmetless, and Brad Marsh recalls that he received a lot more "dirty shots, elbows, high sticks, hits from behind" during the half-season of his career that he did wear a helmet). If, besides considering it unnecessary, players think that wearing a helmet will actually reduce their effectiveness, then that's going to put a drag on voluntary adoption.

Other answers cover a lot of ground, but I wanted to note that the game today is different than the game in the helmetless era. Consider:

• Bernie Geoffrion is given credit for inventing the slapshot, and his playing career was from 1950-1968. Slapshots are much faster than other shots.
• Bobby Hull and Stan Mikita are credited with curving the blades on their sticks in the 1960s, something that enabled harder and more unpredictable shooting.
• Blocked shots weren't a tracked statistic until 2005. Better protection has enabled it to be a viable option.
• Sticks used to be solid wood, but now are flexible composites that produce harder shots.
• Goaltenders used to play standing up, with the idea that their face would stay above the crossbar.

It was dangerous to play without a helmet, but not as dangerous as it would to play without a helmet in the modern era.