My recollection of both crashes was that the pilots hadn't even been informed that MCAS existed, so they were suddenly presented with an extreme repeating nose-down trim and no reason why it would be the case.

That's largely true for the first accident. And yet despite that, the Lion Air captain was able to fly the plane for 5 minutes without any serious issues. He simply overrode MCAS each time it kicked in. It was only when he handed over control to the FO that things went wrong. The FO repeatedly made small trim inputs which allowed MCAS to reactivate, but he allowed MCAS to run for the full 10 seconds each time, and he never brought the aircraft back into trim. The captain then ignored him when he said he was having difficulty controlling the airplane.

Why did the captain hand over control in the first place? Well, he had asked the FO to perform the unreliable airspeed memory items. Those are steps that the pilots are expected to have memorized. The FO had no idea what they were, as if he'd never heard the term "unreliable airspeed" before, which is unthinkable. He spent 5 minutes looking for the checklist in the QRH. The captain finally gave up and decided to look for the checklist himself, and that's why he handed the controls to the FO. The FO's lack of knowledge of critical information demonstrates a massive failure of the airline's training program. Additionally, the reason the AOA sensor was malfunctioning in the first place was multiple egregious instances of improper maintenance, any one of which would have been utterly unthinkable at any decent airline.

For the second accident, the pilots were absolutely informed, as I mentioned already. There was an AD requiring airlines to send the information to all their pilots. For Ethiopian, that info would've been sent via an iPad app. It was confirmed that both pilots had at least viewed it, but of course there's no way to know if they actually bothered to read it. Despite having access to that info, the captain proceeded to do the exact same thing the Lion Air FO had done. That demonstrates that simply having information about MCAS doesn't ensure that they'll handle the issue correctly. There were serious fundamental training deficiencies at both airlines, and both airlines already had a history of accidents with poor training as the major factor.

Furthermore, my recollection is after the first crash it was found that MCAS was only operating on one AoA sensor, so when that sensor fails MCAS may do bad things.

Not sure what you mean by "it was found." Obviously, the people who designed it knew how it worked from the beginning, and so did the FAA and other regulatory agencies. No-one perceived it as a serious safety issue, but they were looking at it with the assumption that the pilots are properly trained.

A software update was developed, but not required to be installed by either Boeing or FAA.

Wrong. A software update was being developed, but it just wasn't ready yet by the time the second accident happened. It had only been 8 months, a very short amount of time to get aircraft software developed and certified. As I mentioned, the FAA estimated the probability of a second accident occuring before the software was ready, and they determined it to be very low. But again, that's assuming well-trained pilots.

This was a 'let's rug-sweep this, we'll install the software quietly and nobody will know how bad it was' fix.

As I already mentioned, everyone involved was kept well informed after the first accident. As members of the general public, you and I weren't privy to every detail. But the people who needed to know absolutely did know. It's well-documented that representatives from Ethiopian Airlines were present in the online meetings with Boeing, and they also sent later follow-up questions by email, which Boeing answered. Whether that info truly made its way to all of the pilots is another question.

The elephant in the room is the fact that the standards for pilot qualifications have been dropping for decades, all over the world. When you combine that with countries that have nearly non-existent regulatory agencies, and airlines that will promote someone with only a few hundred hours of total time to captain, you're going to have issues. This wasn't the first accident for either airline, and sadly it definitely won't be the last. The fundamental problems haven't been fixed because it's easier to simply put the blame on the big, evil, American mega-corporation.