Also, I doubt President Shinra would care about the particular details of how SOLDIER works. He would not be abreast in real time of every development, especially since it doesn't directly involve profits. Look at how he treated Hojo in that executive meeting. He didn't care about the specifics of how Hojo would get what he wanted done. He just wanted results.
He's not the kind of person who cares about lore details.
You really may be right about this, but I find that to be such an obtuse way of writing the character, I really hope you're wrong. Shinra is a megalomaniac. SOLDIER has had numerous problems including the mass desertion that threatened Shinra's very existence. If the President continues, at least by then, to ignore oversight and watching the program, he's a complete and utter idiot that shouldn't have been able to build his empire.
I feel there's a pretty big gap between understandably failing to connect him to the escapees from Nibelheim that year and deciding he's part of Genesis's troops deployed to Wutai 7 years ago. Cloud's not that old and quite crucially does not look physically identical to Genesis. Also the President says "this is classified off course but all SOLDIERs know the truth." To which Cloud replies "yeah". Again, the Genesis incident took place ages ago. There's a whole generation of SOLDIERs that surely joined up after Zack and Sephiroth were classified deceased that have no reason to be clued into this classified information that is no way pertinent to anything presently going on.
The Turks are the big issue here. They've likely seen Cloud, know Cloud was with Zack, and should be able to make that connection
somehow to Zack. Even if it's just the sword. Reno and Rude should be able to make that connection. That sword is too legendary and connected to Angeal to just be on some random SOLDIER's hands. I don't know. This is a weird sort of plothole that I really don't feel comfortable with.
My headcanon solution:
1. The Turks know but they're holding that info internal because they have this weird loyalty/respect for Zack and we'll see that play out in more detail as Remake goes on. Tseng knows, at least, even if Reno and Rude don't.
2. We don't know that ALL Genesis-deserters went through the process, so it could be that throughout the past years, some survived or surfaced, etc. I could see the bulk of Shinra wondering if this is the case for Cloud.
3. No one is freaked out that there's a rogue SOLDIER. This means desertion, leaving Shinra, etc isn't too out of the norm. Probably cases beyond Genesis/retirement, etc. I'm not sure there's anything "forcing" a SOLDIER to remain in Shinra's employ other than the Turks
probably hunt you down. But we do see other SOLDIERs retired and out of service in the compilation (thinking the guy in Nibelheim in Kids are All Right as well as Shopkeep in Junon.
4. The decay/degredation that the President and Elmyra reference is both a holdover from earlier materials (mako addiction) and a process that's probably just not yet touched upon. We don't yet see many
old SOLDIERs, and most that go through active duty meet a crazy end that we see. Angeal, Genesis, Sephiroth, and Zack, according to official Shinra files, must point to them all going some manner of insane before their end. SOLDIER has a cost... sanity, life expectancy, potential Mako dependency, and more.