Papa archeron had a broken knee, he wasn't chronically ill or something. He was healthy enough to go out into the town and sell his carvings so no, he didn't need much care. Infact he was the one whom feyre would ask to bring out the knifes when cutting the meat, not nesta or elain. Nesta would literally hide his stick because she thought he was healthy enough to do something but didn't because of his own pride.
You mention washing clothes, sweeping & washing floors. All these things are important yes but who did them? Maybe the father but he was disabled, maybe nesta but she threw a hissy fit when asked to cut some wood and elain is described to be too oblivious to realize that she could get her hands dirty too. Feyre enters their home covered in blood and elain asks her"how long will it take to clean this?" It is very clear that none of them offered any kind of help to feyre. She says she learnt to prepare & harvest her kills thanks to the instructions of others. There is no way nesta or elain knew anything about hunting and cutting up meat so the "others" here is probably the father or the strangers in the village.
Don't you think washing clothes and floors in the cold would make their hands look even a little bit chapped? But Nesta's hands and nails are described to be neat and elain's skin porcelain and unscarred.
They didn't have any vegetables or spices to cook, they couldn't even afford tea. It was hot water they would drink along with the dried meat and it was feyre who would prepare that. So idk about the cooking part. We hear elain can garden in the first book but nothing was ever said about her cooking. She probably learnt baking in acowar with nuala & cerridwen (she might have developed an interest in that when they had gotten wealthy again because they couldn't afford that in the cabin).
So no, I don't believe anyone of them contributed much to the household duties without being asked or begged to because nothing in book 1 showed that. Everything, infact points towards them just sitting on their asses.
I've seen some people who actually believe that nesta was doing all the household tasks including sewing, washing and etc etc when nothing in the actual book supports that. It's all people's hcs. Sjm didn't go into every minor detail of their life in the cabin but the few things she did show us in their life in the cabin, all tell us one thing: they were all useless.
Just adding here that I don't hate Elain but I admit that she wasn't exactly a great sister to feyre in the village. The sisters are flawed and that's a part of their journey, erasing that just takes away from their growth.