I suppose it's sentimentality. If you watch TNG (which is difficult nowadays -- it's so far out of time), the writers made some interesting observations.
We now have the capability to ask the computer for information (Google), the ability to send each other messages (email), and we can verbally ask the computer questions (Siri). Touch interface (iPhone). These are landmark achievements, and they were all predicted by a show which ended in 1994, before the advent of the internet. The technology predictions have generally been on-point, and it seems like a useful metric for how far humanity can progress.
(To those who haven't watched TNG, the whole series is on Netflix, mostly (entirely?) in the remastered edition, so no TV-like quality. A piece of caution: this is not your typical sci-fi; it's much more positive about the future, science and technology than what we get served on TV these days.)
RE the yardstick thing - there's still lots of things that in Star Trek are shown as easy, and that they feel like they should be easy with sufficiently advanced tech, and that are still hard for us today. So it's maybe less of an yardstick and more of a goal / reminder that there's lots to be done yet.
Simultaneously it postulated that the humans of the future would not be cyborgs, which appears unlikely. Furthermore, few tasks were automated in a manner we might see in our lifetime, let alone the time of the show.
I'd wager most of the things that you are quoting were less pioneered by Star Trek, and moreso popularized to an audience. In particular, using a computer to find information, talking to a computer, and sending messages electronically have long been staples in science fiction. Before science fiction, the only thing that was different was the "electronically" part. And really, I don't know as even Star Trek was treating "electronically" any different than earlier stories used "magically."
Umm. I was reading about TNG on the web in 1994. The internet had been around for a long time before the web was useful to ordinary people. Email has been around since the 70s. Search engines were certainly limited, but were already available in the early 90s.
All these predictions succeeded because they were "predicting" incremental improvement on something already known to exist.
To that end, I'm curious how Star Trek is a useful iconic yardstick? My assertion would be that it is not. It is a common one. But there is nothing useful in it. Precisely because you have to cherry pick what parts you use and how you interpret them. For example, you interpreted it to refer to the method of curing. The comment was taking it for the level of the sickness that was cured. Both are valid places to look.