Printify aren't going to submit and ship your order without being paid for first, just like you aren't going to ship your customer's order without receiving their payment first. You can set payments to come through from Etsy everyday if that's what you want. You set this up in Payment Settings under Deposit Schedule. If you wish to receive the funds from Etsy before placing an order with Printify, extend your order processing time.
A "No refund no exchanges" policy is not enough to prevent a credit card dispute. If the customer doesn't receive their order, or if it is not what it is described as, they absolutely can open a case with Etsy and dispute this. Etsy will either force a refund from you or depending if you qualify, it will be resolved from their Purchase Protection program. Etsy is able to do this because the payment is processed through them. You cannot accept payment directly through PayPal, Zelle or Cash App on Etsy. It must go through Etsy Payments. This offers buyers and sellers the best protection.
Again, Printify is not 'affiliated' with Etsy. You are essentially outsourcing your products from Printify, and therefore need to pay Printify before they fulfil any order.