So, this is the way it works on the stuff that starts with an MK.
In an established cartridge, like 7.62, the operators will identify a shortcoming with the ammunition and or weapon they are using.
This shortcoming becomes a capability gap.
A requirements document is generated (an Initial Capability Document, Capability Development Document, or a Capability Production Document), which is than looked at very hard to determine if the capability already exists in the market, or needs to be developed. These documents are created by the program offices responsible for a given commodity.
Once the capability has been identified as available commercially, or enough R&D has been performed to meet the requirement, the product is procured.
In the case of ammunition, there are a large number of paramters that must be met, including precision from a test barrel. For the sake of the argument, lets say the precision requirement is .5 MOA at 300m.
When the weapon isbeing developed, the precision is defined as "can shoot any given lot of ammo no worse than .25MOA worse than a test barrel firing the same lot at 300m"
Every lot of ammo received is certified, and a certain number of lots are actually tested to ensure the ammo meets the requirement.
A certain number of the rifles are tested to ensure they meet the requirement with pre-qualified ammo.
A rifle like the MK13 (assembled by NSWC Crane) and the .300WM (goes back and forth between Black Hills and Federal) ammo it uses , which have particularity strict requirements, will have every single rifle and every single lot of ammunition tested.
Basically the ammo is made to work in a given weapon, and the weapon is designed to work with a specific ammo. When a new ammo type is introduced, that ammo will have a requirement that it has to shoot reliably in laundry list of weapons, which is validated during procurement.
Nothing in the world shoots current M80 particularly well, because it is, as best, 2MOA ammo from a test barrel. M33 (.50 cal ball) is more like 3-4 MOA ammo. Navy snipers generally use MK211 in their .50s, which are McMillan bolt actions, because the Barretts aren't particularly accurate either.
I went to find a p-spec on fbo.gov, but I couldn't read any of them on this computer. However, if you go to this page, and can manage to open the attachments, you can see the P-spec, and solicitation information.
https://www.fbo.gov/spg/DON/NAVSEA/N00164/N0016408RJN28/listing.html