The important thing when giving people primitives to work with is that they come with time and space guarantees. People are writing low-level code basically in order to be able to explicitly make that choice, rather than having the compiler make it for them.

So, #2 is just a few extra guarantees about the intrinsics of regular machine-words, with minimal overhead for the fast path (even on the slower path of e.g. a 30-bit ring on 32-bit storage, there's still an O(1) set of barrel-shifts and masks that can be used to implement that.)

A fully-general ringed arithmetic value, on the other hand, is more of an ADT. It could certainly be implemented in the same language and act like a number, but it wouldn't be useful as backing storage for other types in the way #1, #2, and #3 are. (Note that #2 is a better backing store than #1 for many use-cases, e.g. encoding IP addresses and subnet masks.)

Time guarantees on a modern out-of-order processor with some complicated cache structure of three or more levels, with virtual memory enabled? Forget it.

Also, if you include bigints in your list of things people want, I do not find it unreasonable to add timestamps (maybe not the best example because of leap seconds) or 'day of the week', 'month of the year' to that list. Most language libraries get those (often including those perfect 86400 second days) before they get bigints.