Integrated LED sconce turns on and then immediately turns off

There are 3 types of LED fixtures:

• Cheap junk. These can actually make sense in some cases, particularly if easy to replace when they die. A significant percentage will die early, but most will last quite a while and the price is right.
• Well-designed from major manufacturers. These make sense for most situations. They might cost on the order of 2x the cheap junk, but they will on average last much longer. More importantly, they are (in my experience) far less likely to have early failure. Anything can be dead-on-arrival - you return it and swap for another. If a light fails after 4 years instead of 5 years, not that big a deal. But if it fails after somewhere between a week and a year, it becomes a royal pain (been there, done that). The major manufacturers are not interested in getting lots of warranty returns and they (generally) build these things to last. (The cheap ones either build a lot of returns into their cost - which tells you the true value of the product! - or disappear before the warranty is up.) This is the best choice for most permanent installations. Unfortunately, I can't post a list of "major manufacturers" here, but it isn't too hard to figure out. If a company has been making light bulbs or electrical equipment for over 100 years, they probably have some idea of how to build things to last.
• Designer/"fancy" stuff. This is where it gets tricky. And this is what you have. In the old days, a designer fixture would still (most of the time...) take Edison base bulbs, either regular base or chandelier base. The rest of the fixture, provided it was built to UL/similar standards really didn't matter. It was a base for a bulb and some wires - all totally standard - inside a fancy case that looked really nice (subjectively) and as long as fire safety was accounted for (i.e., not enclosed too much, made from metal/glass, etc.) you could make it look like anything and, after initial installation, there was never any need to go back to the manufacturer. Even if the bulb base or wires went bad somehow, they could usually be easily swapped.

Not any more! It is more efficient, and provides more design options too (since you don't have to provide as much space for actual bulbs) to include integrated LEDs. That includes a driver circuit (which converts from 120V AC to low-voltage DC) and the actual LEDs. There are electronic components, small wires, etc. that can all go bad. Done right, a good LED fixture will last for many years. Ideally, by the time the lights stop working you are ready to replace the fixture anyway for a new design, because you can't just replace the bulbs. The problem is that sometimes the fancy designers are not lighting or electronics experts. That didn't matter in the old days. It does matter now.

My hunch is that the fixtures were designed with relatively low-quality driver circuits. Over time, they fail. And since they are the same model, they fail in the same way.

Replace the Fixtures

Anything else - trying to tear them down to find loose wires, replacing driver circuits, etc. - will be a waste of time/money.