I spent a year at the South Pole, where Iridium was, practically speaking, our only link to the outside world for most of the day. The higher bandwidth satellites are only available part of the day, either due to their orbit putting them below the horizon (Iridium uses polar orbits, unlike most communications satellites), or them being booked by other users. As you say, it's priceless for the occasional problem where help is needed from experts up North. Nominally, the station also has 24/7 text email through Iridium, and a couple voice lines, which is mostly a good thing for people's sanity.
Shortwave radio could address some of those uses, and there is a one there too, but it would be a rather different experience without something like Iridium.
the existing starlink test satellites are in highly inclined orbits, so if there were a great deal more of them, this would also be a possible solution.
The original 1998-2000 marketing materials for Iridium are ridiculous, like a traveling executive is going to take a gargantuan expensive satellite phone with them when traveling internationally, rather than relying on GSM roaming agreements.