OP, I'm curious, do you have children? I just wondered because a lot of your responses suggest very little experience of baby and toddler related topics and seem to be based on a very theoretical idea of what parenthood is like. I also remember assuming that I would stop breastfeeding when the child had teeth, which I assumed happened at some point before the first birthday and I also thought that stopping breastfeeding and starting food was something that happened all at once, I didn't realise that both usually continue alongside each other for several months (if not, indeed, years).
As several people have pointed out, BLW usually refers to a process of introducing solids in which you let the baby set the pace and usually give them pieces of ordinary food rather than mushing food into a puree and spoon feeding it to them according to a parent-led plan or schedule.
Stopping breastfeeding when the child wants to stop as opposed to the mother deciding is usually called self-weaning.
Breastmilk supply does not usually just randomly dry up. This can happen during the early weeks if supply has not been sufficiently established for various reasons (not all under mum's control) but it doesn't usually happen later on, basically as long as you keep feeding you can keep producing milk near indefinitely. In fact once supply has been established for a while it is remarkably elastic, you can stop feeding for several days if not weeks and still produce milk. It is also possible for a woman who has previously breastfed to induce breastfeeding again later on, apparently even post menopause. This was apparently done in many societies before formula milk was commonly available, for the purposes of sharing childcare. Many cultures have a word similar to "milk-sibling" meaning a person (usually a cousin or other relative) who was breastfed by the same woman, and there is a taboo in many of these cultures against future sexual contact/marriage between milk-siblings. It can be the case that the flow slows down and isn't particularly interesting to the child any more.
Women have many and varied reasons for choosing breastfeeding and it is not always to do with the health benefits, although this is often the reason that people automatically cite first, so it tends to show up as the most common reason on surveys. I think this is more of a cultural thing - it shows you what we value - we are supposed to make choices in parenting which are selfless and which benefit the child. Research shows that the most influential factor is generally what your own mother did. We want to do what feels familiar and normal to us, particularly in such an intimate act as feeding a baby.
I certainly never really found it unpleasant or annoying (except for the pregnancy related aversion) - and teeth are not an issue. Breastfeeding releases oxytocin which is a relaxing and feel-good hormone, so it can definitely feel good.
Nobody ever explains the "mothers are doing it for themselves" comment so my only assumption is that it is either so deeply cultural people who say it don't really have a reasoned understanding of what they mean, it is more just a feeling of going against the "mothers are supposed to make selfless choices in the interest of the child only" cultural value. Or they mean that the mother is infantilising the child, which only makes sense if you assume breastfeeding is only for tiny babies, which they often do think, so this makes sense (even though I totally disagree with it). And it's just nonsense anyway, I mean look at the overall parenting to see if she is infantilising or not, don't single breastfeeding out, it doesn't make sense. Or they mean that the mother is getting some kind of sexual thrill from breastfeeding, which is so disturbing and odd that I can only guess they are closet breastfeeding fetishists themselves or something.
I did EC with DS1 as well (but not the others). Ask me anything
it is definitely not "just letting them poo on the floor" but poo on the floor happens, inevitably, at some point with young children just like it does with puppies.