Yes, and as you suggest LBC and Capital were pretty much the nearest thing we had to national independent radio in those days - there was a glamour about them because in the days before national radio, they were the only big mainstream alternatives to the Beeb. Indeed local stations flourished in the main probably because we didn't have national radio at the time.
Like others on this thread, I'd have loved to have been in London when Capital and LBC and LWT and so on were in their pomp, because elsewhere in the UK, with its single one-size-fits-all ILR station and dull regional news, they always seemed incredibly exciting, but that's mostly because they had a massive amount of glamour connected to them for being quasi-national affairs. They certainly did stuff for London but as far as the audience was concerned I doubt that was the attraction.
It also helped that Capital was on FM at a time when Radio 1 was mostly medium-wave only. I grew up in London at the time, and Capital was everywhere. Radio 1 was something you listened to on holiday. And the original LBC was a unique format, largely inspiring Five Live (indeed, Brian Hayes - very much the voice of LBC during the 80s - went to Five Live).
I was being a bit whimsical, but I do think the value of local news is overstated. I watch the local news, but as a supplement to the national news, not instead of it. When I get up in the morning I want the national news, local news is simply not important, unless you go really hyper-local and talk about bins.
The rest of it just doesn't really matter. It's like some of our American friends on this forum referring to "pulling up the radar" to check the weather. It doesn't matter in Britain, a single forecast is enough, the weather is barely that bad to have any great impact. The tube being down is an inconvenience, yes, but I can sort that out myself.
I take your point, and I don't blame you for saying so. But there is the public service angle here - local media scrutiny of official bodies is at an all-time low, and that's something that's been highlighted by the Grenfell Tower tragedy, where the local paper closed four years ago and the local council and its attitude to its housing stock had been pretty much left unreported, except by a residents' blog.
Television isn't the best medium for this kind of coverage, but the papers which feed into London's news coverage are mostly giving up this kind of reporting now - many London boroughs, particularly across south and inner west London, have no investigative journalism outlets, and some haven't had for years. (Indeed, it's now the BBC that's being asked to bail out local news publishers by funding pooled local council reporters.)
If this was 1987 or even 1997 it would be easy to imagine the fears of the Grenfell residents would haven been picked up by a local paper, then picked up by, say, The London Programme, and the relevant decision-makers held to account. That chain has broken now.
This boring stuff does matter - it really can mean life and death. While nobody's suggesting a red button multiscreen open of 32 different council scrutiny panels, the whole system - on TV, in print, on radio and online - is bust. London Live was never the answer because its model assumed things were just fine and dandy. And now they've blown it, and Ofcom let them.