I have noticed the exact same thing. All of the things that other folk see as bad, the bemused looks because she talks excitedly and interrupts them or she wants to do something and I let her and I can see people thinking she cant pull your trolley in the shops etc etc.
She is amazingly well behaved and I think it is exactly as you say above because she is respected and treated as one of us and not just a thing or an annoyance!
I always laugh thinking of all the confused faces and obscurity people think when I'm out in public with kids.
Especially from family, who have many times been holding their kids and then they walk away, and then the kid starts crying (usually baby, obviously) and I quickly run up and request the baby, they give me a weird face, I take the baby to what I think they want and then they're happy and we're back on track wherever in seconds.
Now the obscure face of wondering what I did and how I knew, and why would I go out of my way to do that, all bundled in one expression.
Little do they realize that not only is it good for the baby to do that, but it's good for you as you don't have to have an upset, sad, loud, crying baby.
Silly regular people clearly must not speak baby, or toddler, or maybe they just don't care :(((
Why not let your kid take a trolly in, especially if the store has baskets, it's really not obstructing anything, and besides, nobody really cares. They'd think it's weird, and obscure, but cute~