Extroverted children are exhausting. They get their energy from being around other people. That's where they're relaxed, calm, happy, energized.
I am NOT an extrovert (although I'm not shy, either), but my son is. Older kids, younger kids, adults... it doesn't matter. It's taken YEARS to get 15 minutes "to myself".
The idea that you can teach an extroverted child to play by themselves by just "doing it" is as "simple" and idea of taking an introverted child and just dumping them in a crowded room of running children and expecting them to be happy.
More people have introverted children/ slow to warm up/ shy children than extroverts. It's just a numbers thing. Most parents have to coax their children to interact with others, have to worry about them getting overwhelmed, and go very slowly with the whole process or they just freak out. It takes YEARS. And even then, as teens and adults introverts still make sure they have sufficient time to themselves.
SAME process with extroverts. It's just that you won't have many parents who have the same struggles you do. And it's not 24/7 the way it is with extroverts. Because you have to take BOTH to social situations, but an introverted child will do "quiet time" and "play by themselves" naturally at home. So the "work" is only when you take them somewhere.
Extroverts, the work is constant.
I homeschool my VERY extroverted child. (Oy, vey! But honestly, he gets more social interaction homeschooling than he would awayschooling in our area. Awayschool kids 'round here have VERY short periods of time to talk/play. 30 minutes a day for lunch/recess. The rest of the time they're supposed to be quiet/focused. That doesn't work well. He needs to be INTERACTING with people. His teachers LOVE him -he does tons of classes- but it's becuase we very purposefully sign him up for very interactive and group oriented classes.)
Anyhow...I can literally have him surrounded by people 24/7, or take him to a 4 hour class, or have all the cousins over... and while he's with other people... he's great. 10 minutes after they leave, he's back to hanging out with me. Don't get me wrong, he gets bored with me, but I'm better than nothing! lol It's been a REAL learning experience with him. Because I'm the opposite. I love people, but they exhaust me. I need lots of alone time to relax.
HINT: Extroverts need to be around others, being by themselves is exhausting. If you need time to yourself over the next few years, you'll probably need to either import it (bring kids OR other people over) or export it (classes, playdates, school). You can also "fake" it. This is plugging them into the TV/Computer/Games. Something that is either around pseudopeople (actors), or interactive electronics.
My son is VERY independent. He's not shy, clingy, etc. He's just an extrovert. Which is rare.
I now get about 1 hour a day where my son can be doing his own thing. BUT he still pops in ever 5-20 to "show me something!" (funny, cool, interesting, etc.). This is a huge improvement (he had some big cognitive leaps this year)... because last year I'd FINALLY managed 15 minutes (with him talking almost constantly to me from another room, or popping in every couple minutes to 'show me something cool/ get my opinion/ just "touch base"). HE'S NINE.