The operative language in RFC 1918 is in section 3:
An enterprise that decides to use IP
addresses out of the address space
defined in this document can do so
without any coordination with IANA or
an Internet registry.
The implication here is that addresses in the RFC 1918 pool should not appear in the global default-free zone, which is a property they share with the IPv6 unique local addresses, i.e. fc00::/7, defined in RFC 4193. Where they differ, of course, is that the IPv6 unique local addresses are explicitly marked as global scope and not site-local scope, whereas IPv4 addresses don't have any official notion of addressing scope. In practice, RFC 1918 addresses are often used as if they possess site-local or organization-local scope, but it isn't strictly necessary to do so.
It's also not necessary to use NAT to give privately address hosts access to the public Internet. In theory, the alternative is to assign such hosts additional public interface addresses alongside the private addresses. Source address selection is supposed to work.
Alas, when presented with an existing practice that works, people often don't give a flip about how it holds up in theory.