Im an introvert too. And my son is also a bit of an introvert... But he still craves interaction with other kids. Up until recently (hes 2yrs, 3 months) He was happy to just kind of play along side other kids .. watch them, but not really join in. Since he turned 2? He cant get enough. He goes nutso running with the other kids, with the biggest smile on his face.
He drives me mental at home "Come play with me! Come play with me! Be an airplane! Jump on the bed!! GO GO GO!!!!!" I cant do it. Its so exhausting. I cant be my toddlers only source of entertainment and interaction. No adult can.
Not to say they need to be with kids every day of the week. But I do think it absolutely has its benefits. Despite me being uncomfortable and awkward as balls, I make the effort, for him.
I found a play group, I enrolled him in toddler soccer, I take him to parks or play centres where I know other toddlers will be... And I let him loose. I hover close by, make sure hes not snatching or hurting anyone. I give other parents a nod or a "how olds your little one?". Even though Im panicking the whole time, I still find its really easy for 2 parents to talk about their kids. So I stick to that. 5-10 min of banter, then I head off to get the toddler some wipes or their drink bottle (while mentally regrouping and calming myself down lol. Yay social anxiety!).
You dont have to be the worlds most sociable mum. Youre under no obligation to make friends with the surrounding parents. If your toddler is naturally social, he'll lead the way. Some days I'll take mine to the park and he'll stick to himself ... other times, he'll run up and start hugging randoms. It also feels like toddlers only have 2 modes when playing with other toddlers...
1). Waving nicely 2). Snatching, pushing or yelling "NO MINE GO AWAY!!!!!!"
Honestly, you'll be so busy refereeing their behaviour, you'll rarely have time to even look at anybody else.
Like I said, I think playing with kids their age is great and has lots of benefits. Its hands on real-time learning, entertainment, and keeps us busy. It doesnt have to be every day, and it doesnt have to be for hours at a time... Their tiny toddler brains get bored anyways.