I've only used the buggy maybe twice since DS started walking at 13 months. I do organise trips out based on how far he can manage but there are times he wants to be carried. Sometimes he is tired but mostly he is bored, it's amazing how they have the energy to play in the park once you get there but grumble about walking there. Mine is older now, quite a petite boy so not heavy but I do have a bad back and basically I give him no option but to walk. He has seen me in a great deal of pain unable to bend or barely walk myself when my back has been bad (always after a phase of teething or illness when I have picked him up repeatedly during the day and night or held him for long periods to comfort him) and I've explained to him why I can't pick him up. If he complains whilst walking I will stop, get down to his level and have a little chat and hug then continue, I also suggest a little rest on a bench if there is one, sing marching songs to make it more entertaining, give him the option to run holding hands so we get there faster, or walk slowly if he prefers, etc. I think for a LO it is so comforting to be held in arms and also they are so small they may feel they are missing out on sights and sounds, conversation is easier when in arms rather than being several feet lower down. tbh I don't like to refuse him but I know I have to save myself for when he REALLY needs me to pick him up. I also now have extremely sore tendinitis in my arm (tennis elbow and golf elbow and no I don't play either) from holding him and playing with him so even more reason to reserve picking up for the important and unavoidable times. I think if you are already suffering with back and arm pain you do need to make changes now before it gets worse.
With supermarket shopping we have a routine where DS begins walking, helps to fetch all the fruit and veg and put them in the trolly, very interactive. Then we head to bakery where he also helps until he chooses his snack/lunch from the fresh baked goods (something like a cheese twist, not cakes or cookies) and is lifted up in the trolly to eat whilst I quickly do the central aisles which don't take me that long. I also carry fruit and drink so he is fully fed and watered. When he starts to get a bit fussy he is lifted out and from there on I ask him to help push the trolly and pick up a few items. He gets to play in the toy aisle for several minutes at the end of the shop then it's back in the trolly and head to the check out. After check out and being patient in the trolly he gets 2 or 3 minutes playing on one of those machines like a school bus or ice cream van, the sort you put money in but I don't put money in (!), he just plays make believe on it and I join in. The point here is that the trip is broken up into several different parts so it doesn't all seem like one long slog for him. He also gets to take part rather than just watch, and when he is in the trolly gets the food and lots of chat so he doesn't feel forgotten.
it takes time to establish routines and expectations but I do think you can just tell your LO you need her to walk. Every parent has to do it at some point for some a little sooner for some later.
Good luck!