If language was the issue you find she simply followed the other children and did what they were doing assuming that is what she was supposed to do such as sitting down at the table.

Sounds more to me like the mom is aware that something is just off and she is trying to convince herself and others by what child can do. Since you didn't see the puzzles maybe she just moved from big knobs to small knobs not to the 24 piece puzzles your other 3 year olds can do.

What was the reason for moving from the other daycare. Could it be that the caregiver had concerns mom did not want to hear. If other caregiver spoke only in Chinese by now the child would be almost fluent since she would have learned the language by immersion - if she only picked up a few words that is either a big concern or says that the fact the other provider spoke Chinese is simply an aside of info and child was spoken to in English meaning she is not processing much of what she hears.

What is her vocabulary, speaking like in terms of complexity for her age.