Dallas here from the Shopify Social Care Team. I would love to shed some light on this to clarify for you.
If you're wanting the money from your customers to go right to your PayPal account then you would need to have PayPal set up as your only payment gateway. This way any of the funds from your customers would automatically go to a centralized location. Even if you were to do that though, there would still be a time required to process the funds. That isn't a delay in association with Shopify, but rather it's with the bank or with PayPal depending on what you use to accept money from your customers and where you want the money to go. In order for money from an order to be delivered to an account, it will have to go through a processing delay of about 3-5 days. That will be the case even if you were to just use PayPal.
So to answer your question, you could make PayPal the only gateway available for your customers to use in order to have all your money in one spot to purchase the dropshipping orders, but the funds will still be subject to processing time.
Just to confirm, are you looking to use Shopify without the Shopify Payment Gateway? If you don’t want to use that gateway anymore then you’ll just need to go to Settings > Payments > Shopify Payments > Manage and then when you scroll right to the bottom you’ll be able to see a button that says “Deactivate Shopify Payments.” Can you try those steps and then let me know how it works for you?
Is there a reason in particular that you’re looking to not use Shopify Payments?