It absolutely is, especially for stuff you really care about keeping secret.
Let me clarify this for you. Public key cryptography is used in specific circumstances to meet specific security needs. The specific security provided by public key cryptography is asymmetrical information.
In most use cases, public keys are "key encryption keys" or used for authentication. I cannot think of any program that uses
only public key cryptography for encryption. The cryptosystem probably in widest use, TLS, uses a hybrid cryptosystem instead of pure public key.
Sure, any modern cryptosystem would be nice and ensure that cryptanalysis is practically impossible, but it is impossible to realize these benefits against WWII enemies. And in any case, rubber hose cryptanalysis will beat all of these.
Navajo works because it's fast and decently secure, at least for fine details, but it's by no means perfect.
It is by no means perfect, but please give us a citation for when its imperfect nature actually mattered.
From what I remember, there was no Bletchley Park, or Signal Intelligence Service, or OP-20-G, equivalent in terms of investment. There's no MAGIC on the IJN/IJA's side to give them strategic information.
Before the tape recorder or other similar technologies made audio recording easily portable, how are you going to save verbal ciphertext? How are you going to disseminate this information on frequency from one unit to another?
And sure, US Naval Codes of the time may have been inferior to what we have now (Rijndael/AES, Blowfish/Twofish, Salsa20/ChaCha, etc) but using them, or rederiving them, or reimplementing them is overkill and a waste of money and more importantly, the very limited mathematician, cryptographer, and cryptanalyst time.
As with previous derails, please cease. There is a time and place for everything, and the time for public key cryptography is definitely not World War II. Give it half a century or so, and then we can talk.