Only the person with the package can prove it was broken/damaged. They need to SEE the package and it's damage. You have it. You have to file the claim. If you ship it back to them... they could file it... then the USPS can say YOU damaged it before sending it back to THEM and deny the claim. Surprisingly it actually makes sense if you think of about it from their standpoint. Hard to accept "Hey, so, uh, the dude that got it says you messed it up". I've heard multiple stories of customers sending the item back in new packaging for the claim.... DENIED. Without the original package at the post office that delivered it (read: screwed the pooch) there is no money.
I've shipped 4,000 USPS packages in the last 2 years. Only 3 claims filed by customers. All paid off within 3 weeks. One was just a shredded empty padded envelope that arrived. Paid off the $100 covered for 'commercial shippers', though normally only $50 on priority for 'standard' customers. It was a $15 part in the package... she may have accidentally handed them the wrong invoice which was for $150 and they paid the $100.
**Edited to add: I have had ~10 'lost' packages in addition to the damaged ones. A few more that took MONTHS to arrive. One I shipped two replacements 1 month apart. They got the third. A total of 4 months after sending the first one the first two arrived within a week of each other. They RTSed both of those for me**
I ship 1 overnight every. single. week. to NY from SoCal. 1 out of 4 times they were a day late through USPS and I get 100% refund. I started shipping a day earlier for the ~25% discount. Now use FedEx since I get STUPID good rates through a private shipper ($21 priority overnight 10:30a delivery for 3lb 10*7*3 box) and they miss it 1 out of 6 times, but are late (time of day) 90% of the time and don't refund, apologize or give a crap if they miss deadlines. Now if it's vital it gets there on time I machine 2ea and ship one set USPS and one FedEx just to be safe.