Not many other events I can think of that have aired on all five. I know there has been a UEFA Cup semi-final on all five - C4 showed one in 1998 as part of their build-up to the World Cup - and I want to say the NBA Finals has also been on them all, but I can't remember ITV showing it at any point.
Yes, the NBA has also been on all five channels. The Beeb showed it in the late eighties, in weekly shows on Saturday nights - https://genome.ch.bbc.co.uk/schedules/bbcone/london/1987-05-02#at-22.30
(they often say that football was such a state at the time that the Beeb showed the NBA rather than Match of the Day, though it actually wasn't during the football season) - and then those programmes were repeated on BBC2 over Christmas. C4 had the rights for a few years in the mid-nineties, and then ITV had the rights, and they did a show on Saturday afternoons called NBA 99/2000/2001 with extended coverage on ITV2. Then when they didn't renew the rights, it went to Channel 5 - I think at that point they had the full set of the US sports, they certainly had the NFL and MLB, and I think the NHL as well.
As you say, C4 showed a UEFA Cup match. This was part of Planet Football, in the build-up to the 1998 World Cup, where each week Steve Cram and Simon O'Brien would look at the football culture in a different country, and alongside the series they showed a few live matches - that, plus the African Nations Cup Final and PSV vs Feyenoord in the Dutch league. But the series was a flop, being dumped from 6pm to a late night slot during the run, and that was the end of that.
Like most 2nd/3rd tier sports - ITV got bored with it, and in soon came back the to the Beeb.
As you say, at the time I remember it being mentioned that one of the reasons the Beeb lost the rights was because the organisers wanted to go in a different, massively more commercial direction. As ever, this was greeted by the press as another nail in the coffin for BBC Sport, the same people who had previously, and then did again when they got the rights back, turned their nose up at the Beeb showing it because it wasn't a big enough event.
ITV Sport has been very inconsistent over the years, as to whether they want to cover many sports like the Beeb, or just concentrate on a couple of key sports. In the mid-nineties, for a while it was pretty much just football and F1 all the way (plus the rugby every four years), but when ITV2 started they were in the market for more sports, and got back into boxing and snooker in a big way, and even did a BDO vs PDC darts tournament one year. But then when the ITV Sport Channel went pop, they didn't need all this content and so most of them were quietly dropped and it was football and F1 all the way again for a few years. And then when ITV4 began they were interested again, and now we have lots of snooker and darts filling hours of that channnel.
In a way, it's the lost era of ITV1's pres, because it wasn't an absolutely terrible idea and concept that would inevitably go nowhere (cf. their replacement), but they were a bit too bland to be any good. It didn't help that this was a particularly dreadful period for ITV, where a lot of shows that were commissioned were either terrible, smashed in the ratings, too reliant on the same few people and usually all of the above. ITV Day was a bad idea right from the off, because they essentially threw away any brand equity they may have had for an instantly forgettable name, linking programmes that were instantly forgettable to begin with.
Yes, I would suggest it was 2006, the early days of the next logo, that were the real dog days for ITV, Simon Shaps was a terrible controller and the summer of 2006 was a disaster, shows were being dropped mid-run left, right and centre, they were relying on ancient repeats to fill gaps (Rising Damp at teatime, Poirot episodes over ten years old at 9pm), the scheduling was all over the shop (with Emmerdale at 10pm for a month which was ludicrous*) and they were doing things like dropping CITV with seemingly no idea of what to put in its place and poaching the Saturday Kitchen team who flopped, and "forgetting" to review Paul O'Grady's contract. They even lost money on the World Cup, and indeed if anything sums up the channel at the time it's the fact they changed their main presenter halfway through the World Cup. A mess of a channel.
* The idea of putting Emmerdale on at a regular slot where it wouldn't get in the way of the World Cup was probably a sensible one, but 10pm was a ridiculous slot. You might be able to get away with it for Corrie, but not Emmerdale, it has a more mature audience and half of them would have been in bed by then.