Yes, insurance claims are potentially easier when the item has been packaged by a FedEx or UPS store. But keep in mind that most UPS stores - at least the ones in the USA - are franchises so they’re not actually owned by UPS. If the people running a given UPS store are unprofessional or clueless, you may end up transferring your problem rather than eliminating it - instead of fighting UPS corporate, you end up arguing with a UPS store owner. So when you choose a UPS store, pay attention to how it’s being run, don’t be afraid to ask questions, see how the people behind the counter respond to being questioned about insurance, liability, etc.
I can tell you from 17 years experience shipping everything from jukeboxes to rare artwork to audio equipment that if you use any of them long enough - FedEx, UPS, USPS - they’ll ruin your week. But no matter who you use, insure with a dedicated 3rd party. We have accounts with ShipSurance and G4S International (which is now owned by ParcelPro). Find a dedicated transit insurance provider who is experienced covering what you want to ship. I’ve had good success with USPS insurance, the few times they’ve been involved, but I would never insure with UPS or FedEx directly.