I wonder if being literally honest but willing to tell Not Technically Lies serves culturally in the same way that being willing to tell "little white lies" but not break an official oath/vow does. People know you're willing to be deceptive, so you're not going to be cornered into spilling any secrets they tell you, you're not going to make a tactless ass of yourself, etc. But they also know that they can trust you if they're willing to spend the time and effort to corner you on anything that's exceptionally important to them.

The NTL restriction is stronger: people don't usually insist that you take oaths, whereas people often accidentally phrase questions in such a way that it's not possible to give a deceptive answer undetected. But I think its modern popularity is because it seems more logically objective: "don't deceive by saying things that aren't true" is more cross-cultural than "don't deceive if your hand is on a special book" or "don't deceive unless your fingers are crossed" or "don't deceive unless you didn't say 'I swear' first".