I think if they go down that path, at some point Square becomes PayPal and is crushed under the weight of trying to do too much.
Square has maintained a fairly narrow focus in terms of clients: small-scale retail outlets. With Square Market they don't seem to be targeting Joe's Mail-Order Bait Shop.... they're giving their existing storefronts a way to sell their stuff online.
I think both Square and Stripe are examples of what great things can come when you take on a fairly narrow slice of the payments market. And PayPal is becoming the cautionary tale of what happens when you try to be everything to all people.
Seeing that Square recently launched a p2p payment system, I disagree with you that Square plans to stay within their narrow slice of the market. I think it would look much more like PayPal (a "Pay with Square" button) than Stripe, but who knows.
I'm sure Stripe, Braintree, Venmo, PayPal and the others are paying very close attention to Square.
