I do grocery shopping for a week's supply on a bike and a large cargo backpack [1], you just need a large backpack to enable most of these "too much to do on a bike" situations. I have carried even some small furniture like coffee/side tables with it. Ceiling and standing lamps as well.

If you don't want the weight on your back, paniers are your friend.

[1] https://chromeindustries.com/products/barrage-pro-backpack

I've got a 3-seater bakfiets and it will take a weeks worth of groceries for a large family.

I do all my grocery shopping online - I have not been to a supermarket in years. I can specify time between 6am and 11pm and have it delivered at that time, for free (there is some minimal amount I have to buy to get free delivery but it is so small that I always get over). I am one of those people who never owned a car and grocery shopping was a bit inconvenient years ago (although for me mostly because I hate standing in lines even if it's just for 45 seconds - there was always some decent store close enough even before online grocery shopping was a thing). Nowadays I would not bother going to the store even if I had a car that runs for free.

A total non-issue in most cities where supermarkets are never far. Temperate climate, so rain is hardly an impediment; just dress accordingly.