You use Receive Payment, because you need to show an invoice is Paid.
You don't need to use Invoice when the date of the sale is also the date of the payment; that is Sales Receipt as One Step. Invoice and Payment are Two Steps, to allow for Date differences: invoice = the Sales date and Payment = the Paid date.
And yes, using a Bank account to manage PP is perfect. As you noted, that is where the money went, first. You also need to Reconcile that account in QB to the PP balances from the statement or transactions reporting from PP.
And moving your money is never Income, again. The Sale made it income. Your "from Account for the Checking download is the PP bank. Or, make your own transfer entry, first, as @Kristine Mae Noted. Either method is fine: between these two Bank Type accounts. Not income or expense, again.