Yep, 240 daily departures is about right, 8-14F is easy too because of their consistent fleet. They've also got load factors around 77% [1].
The first class terminal is inside Concourse A (and IIRC, it's a long way to any other concourses). There's 12 gates at Concourse A IIRC (each with direct boarding from the lounge). Assuming each is an A380 with 12F, that's a peak of 144 people, or right around 110 assuming F load factors are consistent with their fleet average (although I seriously doubt it).
It's napkin math obviously, without access to internal data. It's also not so easy as averaging because Emirates operates a partially banked hub at Dubai so there's a whole bunch of arrivals and departures clustered together (check their early morning departures).
The "not having to see/deal with other people while you're waited on" is the value proposition of the DXB F lounge. Each of the top-tier global airlines has a different service philosophy for catering to high-value pax. For instance, Singapore waits on you hand and foot hoping to anticipate anything you could want. Cathay Pacific on the other hand FAs tend to leave you alone assuming you value your privacy, and you're supposed to use the call button (which is basically a starters pistol -- seriously, they'll be there in 7 seconds flat).
[1] https://www.emirates.com/media-centre/emirates-group-announc...