I had a period of behind effectively a stay at home dad and I disagree with this completely.
Being a stay at home parent doesn’t literally mean you have to stay at home. Take the kids and leave the house. Go on adventures. I met so many people randomly during that time.
It was vastly more social than sitting in an office or working from home alone.
Did you struggle with dirty looks at the park?
I wasn't a SAHP but I'd spend time with my kids at a park nearby and people would give me dirty looks for playing with my kids if my wife wasn't present.