The major point of complexity I know of is that most modern washers, at least all of the HE ones, effectively weigh the contents based on motor current and use that to determine water fill levels. This quickly turns into kind of a complex thing that requires a computer.

That said I don't know that complexity is really getting that out of hand. I have a fairly modern Whirlpool Duet, an American-made washer although its propensity for playing little tunes makes me think it's an Asian design or at least heavily inspired by the Asian market. It's getting old enough and gets rough enough use that I've had to take it apart a few times for various reasons. It's not super simple, but it's not really very complex either. Some things stand out as improvements. For example, most older washers used gearboxes and belts for power transmission, something that was called out as a strong point in Maytag machines. But this Whirlpool has a large brushless motor mounted directly on the drum, no belt or anything. That seems like it ought to have very few opportunities for failure, and it's also something that would have been hard to do until recently due to brushless motors requiring electronic control. Besides that there is a drain pump motor, water solenoid valves, and maybe five different sensors. The service manual, taped inside the top cover as is common on these appliances, has a one-page block diagram of the whole works that's pretty easy to follow. The sensors do introduce a failure point that can be a pain but it does show error codes on the display and the service manual gives a fairly 1:1 mapping of error codes to which sensor is causing a problem.

One point that has been fiddly and lead to taking it apart twice is the "air dome" which is used to trap an air bubble above some water which is then piped to a pressure transducer and used to measure the water level in the drum. It can easily get clogged up, and you have to do a fair amount of disassembly to get it out and clean it. Whirlpool put out a service bulletin on this and I ended up replacing the air dome and the pressure transducer as well for good measure. That seems to have fixed the problem and the good news is that the parts were rather inexpensive, but the labor was a hassle. I have wondered if it would be better off with something simpler like a float switch or optical water contact switch, but those would probably be just as prone to getting fowled up if not more.

I guess I'm a little skeptical that modern appliances are actually that much harder to work on. If anything they are often mechanically simpler than older ones, but electronically more complex. The main complaint is that the control board becomes a failure point and is hard to diagnose and expensive to replace, but I've never personally had to replace the control board in an appliance. Most of the work I've had to do on my appliances, which are all getting up there in years but still working fine, has been replacing pump motors, heating elements, and cleaning things out. Those are all things you would have run into on older designs as well.

The onboard diagnostics in modern appliances can be really great too - when the heating element in my dryer gave out, it started flashing an error code that the service manual just told me was heating element open circuit. On older machines I would have probably had to run it with covers removed to measure current on the heating element. The control board can do a lot of the diagnostics work for you.

Actually, IMO, the worst part of serviceability for front-loading HE washing machines is that they usually use concrete weights on the drum for vibration dampening. These make them a huge pain to move and work on. If you ever find yourself repairing your washer I'd recommend grabbing a socket set and removing those weights as a first step, it's worth the time to take them on/off just to make the machine 50 lbs lighter for moving it around.