A Puzzling Thought from Reb Akiva Eiger on Mashiv HaRuach

OC 114:5

בימות הגשמים אם לא אמר מוריד הגשם מחזירין אותו.

Reb Akiva Eiger there, sk 5

מחזירים אותו. נלע"ד בליל שבת אם שכח לומר משיב הרוח דאין מחזירי' דהא אם התפלל רק מעין שבע יצא כדאית' בסי' רס"ח במג"א סקט"ו אף בברכות מעין שבע אינו מזכיר גשם א"כ במה שהתפלל סדר התפלה כראוי רק שלא הזכיר גשם הא לא גרע ממעין שבע וצל"ע לדינא:

So, he says it may be that if you forget Mashiv HaRuach Friday night, you don't repeat Shmoneh Esrei. Proof - since you can be yotzei Shmoneh Esrei Friday night by listening to the Mei'ein Sheva, and the Mei'ein Sheva does not mention Mashiv HaRuach, you see that Mashiv Haruach is not essential for the nusach of Shmoneh Esrei Friday night.  (I've seen this brought lehalacha, because Reb Akiva Eiger's צריך לי עיון is plenty to create a safek brachos.)

My question: Why would this be true? Why would missing Mashiv HaRuach for six months mean that you weren't yotzei Shmoneh Esrei, and punkt Friday night it doesn't matter.

The truth is, it's not really a kashe on Reb Akiva Eiger. Once Reb Akiva Eiger pointed out that Mei'ein Sheva does not include Mashiv, and that you're yotzei with Mei'ein Sheva itself, then the question is, if Mei'ein Sheva is really a tefilla gemura, why doesn't it contain Mashiv Haruach? It should!

(Please do not tell me "But what about הביננו?" Havineinu is said after the full nusach of the first three, including Mashiv, and followed by the full nusach of the last three.)

Maybe there's something different about Friday night, but I doubt it, because aderaba, rainfall davka Friday (and Wednesday) night is most propitious. Taanis 23 - 

גשמיכם בעתם בלילי רביעיות ובלילי שבתות, שכן מצינו בימי שמעון בן שטח שירדו להם גשמים בלילי רביעיות ובלילי שבתות עד שנעשו חטים ככליות ושעורים כגרעיני זיתים ועדשים כדינרי זהב 

Maybe there's something about ויכולו that alludes to Geshem. If there is, I haven't found it. Yes, the following pesukim mention that no rain had fallen before Adam davenned, but that is not in Vayechulu.

So I'm stuck. I'd appreciate your thoughts.

UPDATE:

The Reb Akiva Eiger is really not well known, so it was a real surprise when Rav Bukspan came up with a tshuva from Rav Wosner exactly on this topic. 

Here is the question.

Someone mistakenly said Ha'el hakadosh in his quiet Shmoneh Esrei, and it was suggested that he doesn't have to repeat because of Reb Akiva Eiger. 

As far as Hamelech, as Rav Wosner says, Reb Akiva Eiger's raya for Geshem is not fully applicable to the question of Hamelech. It is true that Reb Akiva Eiger's opinion in 583:3 is that if you say Ha'eil instead of Hamelech in מעין שבע you do not have to repeat it, but mei'ikar hadin you are supposed to say Hamelech, unlike Geshem, which does not appear there at all. So even if you wouldn't have to repeat מעין שבע if you say Ha'Eil hakadosh, it could be this is because there is a kullah in מעין שבע to the extent of not requiring repetition bedieved, but that does not apply to the personal Shmoneh Esrei.  Whereas the fact that Geshem is not there at all indicates that mentioning Geshem is not important on Friday night, not even in the personal Tefilla.

Rav Wosner also addresses the question of Geshem. He says that first, Reb Akiva Eiger's idea is very hard to accept - it is a puzzlement.  He is docheh the raya;  and I have to say that his dichui of Reb Akiva Eiger's raya is not very convincing. Who cares that this is the nusach? הא גופא קשה, why is the nusach like that?? Then he says that it seems from the Biur Halacha at the end of רס"ח that he holds not like Reb Akiva Eiger.  

He paskens that you can not rely on Reb Akiva Eiger, certainly not on Hamelech, but also not on Geshem, and you have to repeat Shmoneh Esrei.