Liberals and Neo-liberals (what is being called "the left" here) parted ways in the early Industrial Revolution. That was the last time capitalism was left completely free. That was the gilded age, when you had a handful of factory owners working a few days a month and making more than the entire GDP of entire nations, while hiring kids as young as 8 to work 80 hour weeks in factories and mines with dangerous equipment and chemicals, with no liability or responsibility for their safety.
The situation, left alone and free to itself as the classical liberals were urging, was worsening, not improving. In Europe particularly, which in countries like the UK the industrial revolution was further along than the US, workers and the public at large were beginning to rebel. Marx had prosphecied that capitalism was going to mature and eventually that's when the workers would undergo a violent revolution and take over the means of production. That was what was starting to happen. Violent, radical Marxist parties were starting to win elections and doing very well in poll after poll. The situation seemed unsustainable.
That's when the new liberals ("neo-liberals") came in. They were not Marxists. They wanted to save capitalism from Marxism. They did not feel that a violent workers' revolution was inevitable as Marx had predicted. They proposed some commonsense regulations like child labor laws, overtime laws, workplace safety laws, antitrust laws, unionization, etc... and the sense of rebellion started to fade. The Marxist parties started to go down in the polls and fade away.
So in that sense, it was the neo- liberals, not the classical ones, who saved capitalism. Left to the classical liberals, Marxism would have had much more of an inroad in western industrial nations in Europe, and possibly even the US.
