This assumes that every sector of Federation space is heavily inhabited with lots of important areas to protect and requires active protection from Starships. I don't think that's a reasonable assumption.
How much more of a discount do you want? I went back and checked my numbers (doubling sector diameter each time) and got:
- Volume of the Federation: (8,000 ly diameter) 268,082,573,141 cubic ly
- Volume of a sector (20 ly dia): 4,188 cubic ly (64,012,075 starships @ 1/sector)
- Volume of a sector (200 ly dia): 4,188,790 cubic ly (64,000 starships @ 1/sector)
- Volume of a sector (400 ly dia): 33,510,321 (8000 ships @ 1/sector)
Using 20 ly diameter sectors, at Warp 9 (a speed most ships cannot maintain indefinitely) it would still take a ship
5 days to cross just 1 sector 1 time. (Not allowing for corrective factors such as subspace sandbars, warp "highways"/"currants", etc.)
A ship assigned a 400 ly sector would take
93 days to traverse it's diameter
once.
When I mentioned the 'only ship in the sector' trope, I was simply pointing out that Starfleet typically doesn't bunch up its ships very much unless it absolutely has to,
Because that would make things
worse in terms of response times. Bunching ships up (as in wartime or to handle busy regions of space) would require even
more starships to ensure adequate coverage.
I did numbers above for a
20 times reduction factor and still got 8,000 starships. Taking travel times into account, I think that's too big of a reduction.
Probably not, but when you consider (again) travel times, you have the Cardassians on the "west coast" of the Federation, and the Klingons/Romulans on the "east coast", with many 100s if not 1000s of light years between them.