Ref # A Dictionary of Advanced Japanese Grammar p315
The use of wa after nakushite indicates that the sentence is a conditional sentence. This means that when a sentence has a conditional meaning, wa is mandatory as shown in 山川教授... = perhaps nakushite is "without" = perhaps nakushitewa is "if it weren't for" - slightly stronger
I think a good translation for this is "If it wasn't for_,".
I agree that it should be translated as " If it wasn't for...". By the way, in ex #5309 the usage of なくして seems strange. Because it sounds strange when translated, [ If it wasn't my permission, you should not have done such a thing]. 〜なくして（は）．．．ない where the usage usually comes with "If it wasn't ..., then..(followed usually by sentence stating : can't / doesn't have the potential or possibility to do or have something)"
It should have been : 「私の許可がなかったら、君はそんなことをすべきではなかったのだが．．．」 If there wasn't any of my permission, you shouldn't have done such a thing...
Or 「私の許可ではなかったら、君はそんなことをすべきではなかったのだが．．．」 If it wasn't of my permission, you shouldn't have done such a thing.
Or another example for なくして, 皆さんの協力なくして、そんなに辛い事を解決できないのだ。
As for #5309, I don't think the Japanese is wrong even if it sounds strange when translated. Please advise a proper translation for it.
ex #5309 would mean that if it wasn't of my permission, you shouldn't have done such a thing.
I am actually doubting the usage of なくして in that example, not the Japanese. Instead I suggest: 私の許可なくして、君はそんなことを達成するのが無理だ。 If it wasn't of my permission, it is impossible you could achieve such a thing.
After なくして、 it is natural to be followed by something that one can't achieve or something that is impossible to happen without it.
In #5309, it is 'shouldn't' instead of 'can't/impossible'
Here are some examples from a book: 1.彼は、人工呼吸器の助けなくしては一日も生きられない体だ。
There is a difference between 'If it wasn't for...' with 'without'.
みきさん、 Below is some explanation from my 先生： 「私の許可なくして＿＿＿」の文は日本語ではあまり使いませんから、 正しい文は「私の許可なしで＿＿＿」でしょう。 or 「私の許可なしに、______」
私の許可なくして is perfectly valid in terms of written Japanese. In terms of spoken Japanese, perhaps 私の許可を受けないで would be normal, even though 私の許可なしに would also be acceptable.