The part where you said "The only reason why we even know they did this is because of the parser." when you were stating some types of bad performance that could easily be spotted without a parser.
Not sure why you're saying it's difficult to say what went wrong when you said the paladin did shield lob spam, and the whm spammed cure 1. Shield lob has a pretty distinctive animation. You only need to look at the party ui to not only see the cure 1 spam but also the most likely very poor state of party hp.
I'm talking in general. As a healer I'm always looking at the party ui so I can spot a lot of things pretty quickly such lack of buffs, lack of or poorly timed defensive cds, poor or incorrect choice in hardcasts, etc. The tanks in my fc can spot bad tanks super fast just by watching their skill animations. You might find it hard to believe but you can actually see visual signs of people not understanding their class without a parser.
I always pay attention to what my co-healer is doing as well, especially in pugs because some really don't know what they're doing. Maybe I'm weird for this but I'd rather try to carry a less skilled healer and try to get a kill, than waste time wiping because I didn't bother to adjust to the situation by ignoring what the party ui can tell me about them. And yes this does include savage in the rare times when we need to pug a healer.
Personally I feel parsers truly shine when people are trying to get the most out of their class or team. The results of small changes in stat weights, openers, alignment of party buffs, etc, can be incredibly difficult and often near-impossible to track adequately without a parser or fflogs. You can to some degree see changes with your own eyes but without a parser you can't see if a particular use of skills worked better on that pull because it is better or because the players were very lucky with crits and dh. To get the real truth from this you need something like a parser or fflogs.