In several prior jobs we shipped tens of thousands of boxes a week, mostly UPS, some fedex, and a few USPS (including the hybrid services) as well as LTL and FTL.
USPS by far lost the most items. UPS and Fedex would very, very occasionally lose one, but rarely. Most "lost" actually showed up later, having been damaged and rerouted to figure out for whom it was intended, e.g. from internal contents.
Even though USPS was cheapest, we used them only for tiny business-to-consumer items, like individual books or CD's. They are just not as reliable.
Can't comment on insurance claims, we self insured everything. But if I had to ship and expect it to arrive, I'd ship UPS (ground or air) with signature required. If I also wanted it to get the most gentle treatment I would ship Fedex air (not ground).
Another option that is very safe is both will allow you to ship "Hold for pickup" so that it has to be handed over by an actual UPS/Fedex employee (ship to a real UPS/Fedex facility -- (almost) none of the packing stores are owned by them nor are they real employees of the carrier, they are franchises). It travels in sealed and locked containers between facilities (no driver access), and never goes out on a route truck alone with a driver out of video range, nor sits on your doorstep waiting for you. Theft is still possible, but a lot less likely.
Good advice. My experience is similar in some regards and different in others. I worked for a company that had one of the highest shipping volumes in the country. We only used UPS and Fedex (every kind of air and ground for both) and Fedex had a 4-5 times higher loss and damage rate than UPS did. Fedex still wasn't what I would call horrible but it was very substantially higher than UPS and as a result we used UPS as often as possible. Fedex was also more expensive about 95% of the time and so that made the decision even easier. This was about 13 years ago so I can't vouch for lately but I don't think a lot has changed. From what I recall that I have read in several places that I felt were reliable, and my personal experiences, USPS actually isn't worse than UPS or Fedex and may even be slightly better, but I couldn't swear to it. I also can't speak to how any of them handled insurance claims.
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