Can't provide input on very bulky items, but It seems you don't use panier bags or s front basket. Those helped me do much with most items. Some slightly bulky items as well. Add some old tires tubes (free ones with holes from bicycle shops) and I can hold a lot of things.

An e-bike might be a better option if you have many hills and are carrying a lot of weight, if not a proper e-cargo bike. But for the former you need to be careful about how adaptable it is to racks and fenders etc, since some are not great.

I have a bicycle crate in my rear rack (40L from memory). I can just throw my backpack and/or shopping in there and be on my way. No issues transporting when empty. I avoid riding in the rain but I guess a waterproof bag would help for that. It's durable, the main concern is the rear rack. I had to replace the cheaper rack that I bought last year after the welding snapped in a few places over time (I had it held together with duct tape for a while). My new rack should be much more sturdy this time around.

I have access to borrow a car which I do every few weeks so I don't need to over engineer my bike setup too much.

I've used (and do use) a lot of different things. One of my bikes has a front basket which is very helpful. I have a pair of large panniers that clip on a rack on each side and a smaller one that folds over the top of the rack. I have a long-tail cargo bike which is amazing, it's got almost as much space as a USA shopping cart. I have used a trailer, mostly for pulling kids but they work well for cargo and usually clip on and off quickly and some fold up flat for storage. The long-tail cargo bike is the least appealing if you don't have somewhere at street level to store it, but I think it's the best option if you do.

When I was young and healthy enough to use a bike, I went shopping with a trailer for my bike. It was fully capable of transporting 60 kilograms of stuff. Was a bit tough at the end of the trip when the road went uphill at the end, though.

I'm not car-free, but I do all my grocery shopping without a car. In fact, I'm at my local grocer as I type at 1.7 miles away taking 250 feet elevation gain and 210 feet loss. I understand your pain!

Here's why I still think a bike fits your situation. Namely an inexpensive folding bike with a front basket and rack.

Hills are conquered in the same way as a cart: walking uphill. Also known as "hike a bike." Folding bikes are usually also allowed on busses, so you could take a bus one way. You could time your departure to a bus schedule and shop knowing you don't have to rush or spend a long time at the store.

Folders can be brought inside and consume about the same space as a folding trailer.

My overall point is a folding bike is a trailer that you can ride downhill in. Electric would be a nice upgrade, but it's not necessary.

First the unhelpful answer:

I transport groceries with my e-bike and a bike bag on either side of the rack. They may not be quite as fancy as the ones you can unhook and carry into the store and the design is probably from the early 2000s, but they're BIG, so they're great (apart from the fact that they create so much drag that cycling without the motor is an ordeal).

Other than that, I did use a shopping trolley for a while. It was fine enough. Wouldn't recommend to transport cats in those though, you probably need something with tires, not just wheels.

Most wagons will suck on stairs. Heavier loads will be difficult to manage on hills, both going up and going down. Curbs are usually manageable.

In general it is usually easier to pull than to push.

As long as you have proper gearing for it you can go up almost any incline in a bike, even with a lot of cargo without needing electric assistance. It will just be slow and take a while. I am not a fan of pushing the bike uphill, especially when it is heavily loaded because leaning your whole body against the weight is much more exhausting than just pushing the pedals.

I have a rear rack on my bike and big pannier bags which allow me to carry a 1/2 to 3/4 full shopping cart. When it is fully loaded it makes the bike feel a little weird but it is maneagable and on the flat its not much slower than without them.

I also have a trailer that was once used to haul me as a kid, I use that if I need to carry bulky items or just more stuff in general. I am a bit slower with it, especially because the connecting arm on my trailer is a little bit flexible, so when going uphill and the trailer is heavily loaded it will flex with each pedal stroke and take some energy away.

I dont have an E-Bike because the stores are all closer than 5km around me and I do have some steep hills but also some flat areas, so I dont mind the extra workout.

An old used mountainbike with a wide gear range and a used bike trailer would probably fit into your budget, a lot of the kid/pet trailers can also be pushed/pulled by hand with some quick conversions. And if it isnt that expensive looking you can probably keep it locked outside without anyone stealing it.

If you have a lot of bad sidewalks I would look for wide tires and/or suspension in the trailer, then adjust the tire pressure accoring to the load.