This is Ren from the Shopify Support team, thanks for your post!
That definitely sounds like a frustrating scenario. Unfortunately there is no way to automate the processes of submitting tracking numbers to PayPal. The good news is that this 21 day waiting period is temporary and there are steps you can take to becoming an Established Seller so you can receive PayPal order money right away. My recommendations are to focus on the steps outlined by PayPal to work towards becoming an Established Seller including:
It'd be worth contacting PayPal to see if they can offer specific numbers on the expectations above.
If you have the tracking number on Shopify, the app Add Paypal Tracking Auto will help you update automatically the number to Paypal. It will be easier for you if you have many tracking numbers on your store.
Hope this information will help you.