The Winchester Magnums are not finicky at all, and it has very little to do with throat angle. The angle that matters most is the angle at the start of the rifling. I suggest you look at several reamer diagrams and see what angle the very accurate cartridges use, such as those used in target rifles, PPC's, Lapua & Norma, even the XC line, this should tell you plenty.
It is not throat angle, which is fine, it is the lack of a section of freebore to assure bullet alignment prior to hitting the lands. In front of the case, the 300 WSM's just has a cone into the rifling starting at .3099" diameter.* Pretty much every SAAMI cartridge accepted by SAAMI since the WSM's uses a section of smaller diameter freebore to assure bullet alignment prior to the taper into the rifling. The lack of freebore also limits how far the bullet can extend out of the case before being jammed into the rifling. The WSM design was for maximum case capacity and speed from a short action. Many argue that the SAAMI RSAUM is more accurate, but Remington had to shorten the case a little to use the better chamber design. This cost them a few fps in speed and is largely why they were less popular than the WSM's. Speed was king at the time.
In many newer cartridges (and some old ones), a section of the bullet shank is actually inside the freebore when the round is chambered. In other words, the cartridge is aligned with the bore by the bullet, not the case neck. This helps to ensure that the bullet is straight when the rifling is engraved instead of being slightly angled. Extended freebore allows the bullet to be loaded farther out of the case before it hits the rifling. The RSAUM, RCM, and PRC cartridges (as well as 6.5 Creedmoor and 260 Rem, etc) do this. Even older cartridges like the 7mm Rem Mag and the Weatherby magnums can do this. None of the SAAMI Winchester Magnums or Short Magnums have this type of chamber.
All the WSM target rifles that I have seen use a custom "match style" reamer that has a section of tight freebore prior to the taper into the lands. This has been common for over 20 years. When you seen a comment that a rifle was "throated" for a certain bullet, that means that a section of freebore was added. There are numerous threads on sites like AccurateShooter and Sniper's Hide that debate which improved chamber works better, but no one suggests that the SAAMI chamber is best.
On the Hornady podcasts, they readily admit that it is common knowledge that it is possible to make an improved custom 300 Win Mag with a longer-than-standard action and mag (to take a longer OAL cartridge), a custom chambered barrel and load custom ammo for it. The US Military does it for some sniper rifles. You can't get these rifles or the ammo at your local sporting goods store, though. If you have extra time and money, you can do a lot of interesting things.
The whole point of getting SAAMI approval for their newer cartridges is make this sort thing less expensive and available to the masses with less expensive factory rifles and ammo. Heck, Sportsman Warehouse currently lists twenty 300 PRC rifles normally priced under $1000 and ten more under $1,200.
* FWIW, this is much better than the regular 300 Win Mag that has a cone that starts at .315" dia. I have read many complaints about accuracy issues with this cartridge.
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