Letting them use a buggy at 4 is a good analogy actually.
I think it works well and illustrates both sides - perhaps this is the place where we find common ground?
If you have a 4yo who is often using breastfeeding (or a buggy) in a way that you as the parent feel is probably not quite right but you're giving in for an easy life, then I agree that parent probably needs to work on being more comfortable with giving boundaries. I have been that parent on occasion, and I can recognise that it is a chronic issue in some mother-child relationships. And there's no reason breastfeeding would magically be exempt from that, and so perhaps when some people say they don't think breastfeeding is right after a certain age, they are thinking about this kind of relationship, and assuming that it's a symptom of it, whereas I don't think it always is.
OTOH, some 4yos are more able to walk long distances than others. Some may be hypermobile and need to use a buggy, even for short distances, longer than most children would - this could be compared to a child who is perhaps more emotionally sensitive, has allergies, or needs the regulating effect of BF longer than most children would. (Because BF is not only food, there is an emotional aspect to it as well, there always is, it is not a case of "linking food to comfort", it's to do with the release of oxytocin and is present right from newborn.) OK, that need might be able to be met another way, but it's still a need which the parent has identified and a method of meeting it which works for both. So why stop?
Some children might need due to their parents' preferences or necessity, or perhaps a one off situation like a holiday, big day out, political protest - to walk much further than most 4yos need to walk. In this case allowing them to use a buggy for some of that walking may be a kindness. And likewise some 4yos dealing with difficult life situations or perhaps a lack of nutrition (more likely in poorer countries than the UK) may have a pressing need for BF that other 4yos don't have. Or they may rely on it for one situation only (e.g. bedtime) or want it on bad days but on other days they are just fine without. Many 3yos, 4yos and older are not feeding every single day.
Some children completely eschew buggies at the age of two, insist on walking everywhere, are never tired, and simply don't need it - and that's fine too, if children can reach other developmental milestones early and late and this is not a problem, why not weaning from the breast?
In my experience, it's not as though you pick a date in advance and decide "I'm not going to use the buggy any more from X date" - you generally find there are more and more outings where you don't consider it necessary and it's more hassle to bring it than it's worth, and then one day you realise you haven't used it in ages and you might as well pass it on to your cousin or put it in the loft in case you have any more children later. Letting a toddler self wean is not entirely unlike this! There are times they want it and you say no because it's inconvenient, there are times they want it and you say yes, there are times you think it might be useful but it ends up not being needed, and the times that you both say yes become less and less frequent until one day you just aren't doing it any more.
Sometimes the decision might be a bit more final - you might need to move the buggy on to a relative or just decide that you've had enough and you're going to remove temptation to give in - and again same with weaning, some people decide they've had enough or they want nights back or they need to return to work or want to TTC or something.