I mean… chains *do* stretch… even brand new ones… but not instantly.
And that sprocket may well be junk I personally just can’t tell from those particular pictures. But sprockets aren’t expensive. You can buy them on Amazon for ~8 bucks each if it’s a standard 17mm mount sprocket… so you might as well replace it. But like I said no sprocket is going to destroy a chain instantly unless it’s like… missing teeth or something
So… I’d soak your chain in oil before you mount it. Make sure your oiler is working. And see what you find. And personally I like to keep a spare chain on hand so you might as well get a new chain if you can afford one right now because if your last chain in question has been stretched almost immediately after mounting then that chain is going to be worn unevenly and have weird stretchy spots in it and you’re gonna have to do something about that chain anyway if that’s true. If not you’ve got yourself a spare chain.
But soak it in oil overnight before you throw it on there.