I have done it using only that secret an no others (which includes no secrets in the Fatface room either), but I have not yet done it without that secret. There are too many guards, you don't have enough ammo, and you have no cover (keep mind mind the only health or ammo in E6L9 is in four separate secret areas, aside from ammo dropped by fallen enemies). But I could envision it being remotely possible, albeit insanely hard, to go without the first secret if you were to use extreme patience, precision, and also have simply some good luck. So has anyone done it?
(Nonetheless, despite the fact that I don't see it as being outright impossible, I'm reasonably sure that when ID made the second room the way they did [a huge four door attack], that was meant to be [at least for scratch starters] more or less a dead giveaway that the first room contains a secret. I'd say if you're scratch starting on hard that secret is all but mandatory.)
As a second question, as far as I know, only boss levels (and in particular, those on Episodes 1, 2, and 6) contain "essentially mandatory" secrets, that being secrets that are not in the technical sense absolutely mandatory secrets (like E2L2, E3L8, etc.), but rather secrets in which you would have to do something extreme to survive without. This is considering every level on hard starting from scratch. Using these parameters, do you consider any secrets on non-boss levels to fall into this "essentially mandatory" category, and if so which ones?