I doubt you will buy a "new" washer/dryer that doesn't have modern features. The days of "fixing" many appliances as DIY are gone, unless you have really good knowledge of the intricacies of them.

I don't know what you call "elderly", but I recently bought new Samsung front loaders and built a pedestal for them to sit on. My wife loves them, even with her dementia. I have to label what each switch does with a label maker, but she gets the hang of it. Granted they are "smart" in that they will send a code error to my phone with diagnoses to fix them.

The only problem we have had was she failed to put back in the dryer door filter once and it sucked a sock onto the motor. Luckily it was under warranty. I apologized to the tech for making him come out on a self inflicted thing and he was quite the gentleman, saying he gets calls like it all the time.

 

Look at Lowe's website. They are making washers now that have some features of the old styles. I think the manufacturers finally understand that consumers are sick of underperforming "High Efficiency" washers. They have washers that lets you select water levels and have spiral agitators.
I have a 5 year old Maytag top loader HE washer. I HATE that I can't select the water level. In 5 years, I have replaced the washer's hub twice now. It gets stripped out because the washer tries to wash with as little water as possible. I keep contemplating cutting a hole in the lid so that I can add additional water if necessary.

 

I would check the reviews on bestbuy.com for the various models. I did this before buying our current clothes washer and learned that the front loading washers would not drain completely and so would start to smell. The new efficiency washers do not have the agitator in the center and this results in our clothes lasting much longer. The agitator was tearing up our clothes as material when it is wet loses strength.

The first washer we bought as a replacement was noisy and so a few days later I requested a return. Bestbuy processed the return with no issues at all. They delivered a new model to us a few days later. I have bought appliances from Home Depot and their third party delivery people are terrible and when I have had a problem it has been nearly impossible to reach anyone at Home Depot to resolve the matter. We now only buy from bestbuy.com any new appliances for our house.

 

I don't know the entire market but I think if you are looking for something that will mimic your 20 year old appliances with the mechanical clockwork controller and mechanical switches, you're probably going to be disappointed. I doubt they DIY easily.

I had the same thoughts a few years ago; a top loader with agitator and water level control - just like our old one. It turns out that top loaders are essentially front loaders, but vertical. Front loaders don't wash clothes 'in' water, but 'through' water, but because you are relying on it to gauge how much water to put it, the clothes might not all move through the water properly because they sitting in a heap at the bottom. I always selected 'deep water wash' (the only other option) but was never sure how much more water it put in because the lid locked as soon as the cycle started.

They do spin faster, though; I'll give them that.