AIBU to not want anyone else to pick up my child?

[TwelveTwentyfour]

DH and I recently adopted toddler siblings and they're still attaching to us, having had two traumatic moves already, first from birth parents then from foster carer who they were with over a year, they're rightly unclear on the fact that they're staying with us forever, even if they've heard the words and can even repeat them. The problem I'm having is friends and even total strangers just pick them up and cuddle them without permission. Yesterday the youngest fell and hit his head- not enough to make him cry, he's very rough and tumble- but a guy walking by who clearly saw we were all together stopped down picked him up and cuddled him, HE WASN'T CRYING! I tried to keep my cool and said "pop him down please" and he ARGUED WITH ME! Saying "oh he really took a bump to the head". Another time we were all in the park and I tend to my distance rather than helicopter parent (after the awesomeness of Love and Logic training recommended by adoption agency), and a woman took my child from the top of a climbing frame when she decided he was too small to make it down (he always figures it out in the end) and swung him round making close eye contact smiling and cooing over him. I didn't say anything that time because while horrific she did put him down and he carried on playing and I was too angry and stunned to keep my cool and construct a polite or even coherent sentence. Fallout from this described below.
So my question is: Is this normal? Do strangers do this with other kids too? Mine are unusually cute and affectionate (not biased on this one, it's really becoming an issue ) but that doesn't mean they're teddy bears or dolls for the amusement of strangers. The big problem this causes is attachment confusion. Understandably, since we met them on a Monday and they were living with us by the Sunday. So they'll say at bedtime: "Mummy, is that nice lady from the park going to be my new mummy?" This happens a lot with all sorts of people who have just decided to pick them up and cuddle them! I'm constantly taking them off people. It's getting ridiculous. I can't quarantine them forever, they need to socialise and grow and play.
So...What should I do?! I feel like putting a sticker on their clothes saying "put me the f**k down, I'm not a toy!" Or a PA sign at the entrance to all our park crawl favourites saying "leave other people's children alone unless in mortal danger!"
Rant over, advice gratefully accepted. Xx

That's very shocking

OP, I understand this approach works for your family and I hesitated to post as I know how hard the first few months can be so I don't want to appear critical. If your LA / VA have endorsed this, it may be a bespoke approach. I just wanted to highlight that it may need viewing in line with other research. Dan Hughes equally talks about boundaries and natural consequences but from a much closer perspective both physically and emotionally.

In terms if your initial question, I would follow previous posters advice and ignore parks where other adults are likely to intervene until much later on in your journey together. My son loved swimming, and it was a great opportunity to bond but he homed in on the aquatots leader every time - just couldn't keep his eyes off her. She is a very matronly, down to earth sort and would try and help me with armbands etc. It became clear that she fascinated him and the benefits of swimming together where being lost so we travelled a distance to find somewhere safer for us both. I went back recently and got the same intense staring at her which was easier to break and bring back to me but showed how this still isn't right for us. He has been with me nearly 3 years.

I found a website I had come across previously that have some quote extracts from Foster Cline's books, some as recent as 2004. Extracts pasted below. Bear in mind, the majority of children tested by foster cline and written about are from adoption or foster care and are displaying reactive attachment issues.

When Chris was four years old, he was diagnosed as schizophrenic. When I first saw Chris at age three, he had a vocabulary of less than a dozen nearly incomprehensible words. … Although he flailed and struggled, his therapist, Laurie Smith, was relentless. Holding Chris’s eyes open, she forced eye contact, and stroked his face and gave him unwanted kisses in the midst of his wails. … Being swung in a circle or being held upside down terrified Chris, and he would then reach out for his therapist.… — Conscienceless Acts (1995), pp. 152-153*

In my opinion, one of the best forms of isolation is the blanket wrap. Using this technique, a child is wrapped, arms down in a blanket, and the blanket with the child wrapped is wedged tightly but not uncomfortably between two chairs. This technique is illegal in many states. Making it illegal is a big mistake. The blanket wrap has many advantages over other forms of isolation. — “Corporal Punishment,” Foster Cline’s website (2004, accessed 11 October 2004)

The major task of the second year of life is for the child to learn “Basic German Shepherd.” Children learn to obey “Come, sit, go, no, stay” messages … — Can This Child Be Saved? (1999), p. 32

Joining the children in their misbehavior can be almost as much fun.… ”Oh, it’s time to go to your soccer game? Oh dear! I know I told you that I’d take you. Sorry, I guess I lied about that … ” — Can This Child Be Saved? (1999), pp. 239-242

One parent purposely allowed a child to become lost in a mall, but kept an eye on her as her distress level increased before being “found.” … a recreation of the early distress/relief bonding cycle. — Can This Child Be Saved? (1999), p. 283

I have a very adventurous 2 year old DC who is adopted (but no attachment issues to speak of - placed very young - would never go to another adult willingly).
However the "I can do anything my big sibling who is 3 years older" attitude coupled with to be honest pretty good physical ability (and small for age) means strangers do often think they need to rescue my DC from the top of the climbing frame.
I don't help my DC but I stand nearby encouraging and hands off. If someone else tried to bring them down I'd say "they are absolutely FINE" and if they persisted I'd physically remove them from the adult.
I am very much hands off as I say but usually stand close with arms folded (and words of encouragement making it plain that I am aware and in charge).

I don't pick up my DC every time they fall over even, frankly, if they cry a little, though as they are well known for crying wolf. With a child with more attachment problems though I might be inclined to as they will learn more quickly that you are the one to help them when they have a problem. My older DC (also adopted) has some food issues which we think may relate to v early experiences and we make sure that there is a lot of reassurance around food, probably more than you'd expect for a DC of their age.