I have never bought a pram (although one came with a carseat I had to get in Singapore where my daughter was born) and have never bought a stroller.
I secretly hate prams and strollers, think pram envy is naff and kids often looks so folorn in them low down like that. But I understand why they are necessary and practical for many people.
I have carried my daughter every day in a range of stretchy wraps, non-wovens, and mostly these days, an ergo for front and back carries. She's 20 months and has her daytime nap in the Ergo with either me or DH walking about.
It helps she's on the light side. It also is convenient for me as either me or DH take her out for much of the day and we're in London and our train station has a flight of thirty stairs.
I carry everything I need (a few nappies and wipes, hat, a couple of toys, snacks and picnic stuff in a Longchamps Pliage bag over my shoulder. Helps I b/feed so don't need to hoik bottles. I've even been known to have a rucksack full of shopping on my back as well.
Only time it has been a pain is wearing certain clothes (I'm a dressy person, not a lentil weaver at all) or when I've carried her for very very long walks -- say she's taken an hour to get off for a nap, an hour and a half sleeping and then on top of that quite a while to walk back/stand up on the train. Then I can get painful feet and a sore back, but it's not a bad back.
It just developed this way. I practically carried her throughout her infanthood, then she was walking, then more convenient to keep slinging and seemed pointless to splash out (eBay annoyingly expensive for used strollers, in fact any big ticket baby items)
Yes, if you live in town, don't have a baby that's too heavy, can travel light (we also have a sportscar and everyone was telling us it couldn't be done, we'd have to get something four door) and yuour baby likes it, then yes, completely possible.