I have relevant industry experience, and I would advise anyone to never, ever use Priority or Express Mail for a high-value item, under any circumstances. Even with third-party insurance.
Theft is much more likely than with UPS/FedEx overnight services, and it's nearly impossible to arrange the delivery so that the parcel is on camera for its entire journey, something easily accomplished with some other carriers. And if the package lost, prosecution is almost an impossibility, as is therefore recovery.
In the event of a loss, Postal Inspectors are typically useless for pursuing package theft committed by USPS employees, even with very high value because they live in perpetual fear of the APWU (their explanation, not mine). They also hilariously attempt to pass the investigative buck to local authorities, who rightly note that it was a Federal crime committed by a Federal employee on Federal property.
The Postal Inspectors recently had a high-visibility thread on another site that very much overstated their value, and ever since that thread went up that's been the narrative. It's wholly unjustified.
Registered Mail is the only USPS service worth considering for such items, and it's actually incredibly good, and tightly controlled. It's arguably the most secure service of the major carriers, and probably where that recommendation comes from. As bad as the USPS is at delivering everything else securely, they're excellent at getting Registered Mail items where they belong. Insurance up to $50K can be purchased (and sometimes over that amount).
For someone with no experience doing a one-off, high-value shipment, that is what I would strongly recommend. The insurance also comes without the endless hidden caveats one finds with UPS that requires a complex decision tree to validate, or the BTW-it's-not-actually-insurance aspect of FedEx's declarations (for which they'll happily charge).
Third-party insurance services are worth their (very reasonable) cost, but can be difficult for the average person to secure. Services like Brink's enter the picture when you start talking about moving items above that level.
But Priority/Express is the absolute wild west in terms of secure package delivery. Temporary employees working with no oversight, with a union that will reflexively and aggressively defend people even when it's fully aware it's defending criminals (sound familiar?), and unmotivated inspectors. It's a recipe for high rates of package loss, which is exactly what it has.