Why is it sick? What are you afraid of? I can only go by my own experience which is that my milk, to the best of my knowledge, dried up sometime around 17 weeks of pregnancy. Maybe I am wrong and I still did have milk; my dd has speech problems and could not tell me although I did ask her. My dd nursed until 28 weeks of pregnancy. I was certainly producing colostrum, which is still nutrition, and it was my wish for my child to decide when she was done nursing. Believe me when I say it was uncomfortable and really hurt sometimes, but it was important to her, so I let her decide. I refused to start a pacifier for her at such a late age. I actually did attempt to give her a pacifier at 6 months old when we took her first plane trip, and she spat it out, looking at me like what the heck is that? Children have sucking needs way past infancy, so what is the problem if it is sucking at the breast? I can tell you what the problem is...it is a puritanical streak in our culture where people make judgements that we are sick, digusting, offensive, dirty, -insert your favorite adjective here- for nursing past the age of -insert your comfort level here- months.
Many years ago I was a language teacher. I also taught the culture of the languages. Because some of the traditions were unheard of to my students, they would often say things like, gross, stupid, weird, wrong, offensive, sick, disgusting. I would not stand for that in my classroom. Just because it is not familiar to you or within your comfort level does not make it gross, stupid, werid, wrong, offensive, sick or disgusting. Let's just call it different.
So, if you don't like my extended breastfeeding and doing it in public, just call me different.

Let's have some tolerance for those who do things differently than you would choose.