Well here you go, this is from the US, but it's a report into child car seats and how they performed in crashes from 1980 when they were not standardised and not legally required.
deepblue.lib.umich.edu/bitstream/handle/2027.42/500/44653.0001.001.pdf?sequence=2
There is description of injury and death including an upsetting one where the car seat actually trapped the child in a burning car. Thankfully modern seats are specifically designed to be easy to operate to avoid this problem. But I think the most surprising thing for me about this report was the fact that many of the car seats were reportedly used incorrectly and yet still protected children when the crash was not so catastrophic that nothing would have made any difference.
I wonder which particular child car seat law the OP's acquaintance was talking about, there have been a few.
1988 was the first law about child restraints and stated that they must be used if you had one, which in practice meant that they weren't needed at all. Most car seats at this point had to be permanently attached to the car which meant they were impractical unless your car was solely for family use. Likewise children over 3 had to use seatbelts even in the back, but only if they were installed. (Clunk click every trip originally only referred to front seats as rear seatbelts were less common).
In 1993 the law now stated that all children under the age of 3 must be in child restraints and children up to 14 in seatbelts, although for babies under a year this could be a carrycot. From 1992 rear seatbelts were legally required to be fitted in all cars.
The current laws requiring all children under 12 years or 135cm to use a car seat came in in 2006, but if you look at the BBC news reports from this time, there was quite some controversy over whether to introduce a law. By the early 00s, most parents were using child car seats past the age of 3 which was the legal minimum requirement. They were easily available and convenient to fit and so most people saw the sense in it. The difference then was that most people stopped using booster seats (if they used them at all) by about 5 or 6 years old, or switched to a booster too early from a harnessed seat. Awareness campaigns were trialled first because it was thought that it was better to convince parents of the safety of child seats and let them make their own decision than penalise people who found it inconvenient or had other reasons not to use a child restraint, it was hoped that common sense would persuade most people to use them except in situations where it was very difficult to, but this proved not to be the case and the awareness campaigns were unsuccessful in increasing car seat/booster seat usage (TBH I don't remember ever seeing any of them so perhaps they were just crap?) So the law was brought in and caused much confusion, but car seat use has increased since then.
I would suspect that the "one life saved every year" figure, if it's not been pulled out of the air, refers to this increase in the law from 2006, being the biggest change to car seat laws in the last 20 years, since it mainly covers the category of older children, and many parents were already using child seats for their children anyway.
Here's a modern report with estimates on numbers of lives saved and injuries avoided though I haven't read it. www.roadsafetyobservatory.com/Review/10074