I wonder if you might want to consider a speech and language evaluation? It's hard to say based on age and my general lack of credentials as a speech and language therapist
, but in older kids the backing up and repeating words is referred to as "mazing". If you think that an evaluation might be the way to go, you should check with your local school district before laying any money out (I know that in our district a referral for an evaluation can be done during the early childhood years, and evaluation is free to families).
Does he seem self-conscious about his speech? At his age, adults are more likely to interact verbally, while other kids are interacting physically, with limited demands placed on speech formulation. Perhaps that accounts for part of the difference between his response to other adults, and his response to kids on the playground. It also occurs to me that showing a positive reaction to other kids may feel less risky--he might know you aren't going to leave him alone with other kids, but think that maybe you would leave him with another adult for awhile, so he is less open to adults?
Of course, that doesn't address the other anxiety concerns you've shared, but it might be a place to start. And he IS young, so you may see a change just based on the passage of time. One thought that pops into my head is (if you aren't already) joining something like Music For You, where parents and kids participate in class together. It's fun, doesn't require separation from parent (in fact my DS spent most of each class in my lap) and it would put him in semi-regular contact with other adults who are interacting positively with their own children.
