Depends what you mean by “back in the day”? The first period the BBC had the rights, live coverage was usually part of Sunday Grandstand and was more or less lights out to checkered flag (and might be broken into), Substantial pre and post race coverage only came in when ITV took over the rights in 1997. As such “The Chain” was mainly heard for the evening highlights show and races outside of Grandstand’s afternoon slot.
This sort of thing gets repeated a lot but isn’t very accurate. By the late 1980s and certainly the early 1990s races weren’t interrupted on Sunday Grandstand. Non European races might be highlights only, or have the first part shown as highlights and then joined live later on. Not always the case though, some were live in full even in primetime. Of course in those instances (as well as for the highlights) you’d hear ‘The Chain’.
By 1993 / 1994 even if they were part of Sunday Grandstand there would often be build up to the race, including technical features, and some European races got their own programme on BBC 1 - presumably because there was a longer event on Sunday Grandstand. One race in 1994 (Pacific Grand Prix) wasn’t shown live for some reason, but that was the last time that happened. Some races had a presenter on site - Steve Rider was at Imola when Ayrton Senna was killed- and the BBC were able to cut to their own camera in the pit lane when RAI’s pictures became too graphic.
It took a bit longer for qualifying (outside of the British Grand Prix) to be live, but it was for all the races by 1996. For European races it would often be on BBC 2 and joined 15 minutes in (thanks Open University) but the qualifying format in those days meant you missed very little. For races in other time zones it would be live in full (although you might have the vidiprinter on screen if it was part of Grandstand and the final scores were coming in).
ITV did increase the amount of pre race coverage and did the same for every race as well as being on site for just about all of them. They expanded the technical features and introduced things like the grid walk. This was a big step forward on what UK viewers had seen before, but it wasn’t quite the revolution it’s sometimes made out to be- it was a more consistent approach though.
Not everything got better though (aside from the ads) - it was almost 10 years into their coverage before every qualifying session was shown live. I’m sure there was at least one Japanese Grand Prix where qualifying wasn’t shown until 10pm, some 16 or so hours after it had happened. Then there was the 2000 US Grand Prix that got bumped to ITV2 when comparatively few people had access to that. Primetime races were often hurried off the air as quickly as possible.
When the BBC got the coverage back in 2009 they built on what ITV had done and 2009 to 2012 was the high point of FTA coverage of F1.