Higher resolutions allow you to see more of the game on the screen if the game supports that, which can be quite nice with many games. With other games the amount shown in the viewport is the same but they may use higher detail in the models. Or they might use a combination of the two, it really depends on the specific game in question.
My primary display is a Dell U3011 30" @ 2560x1600 (16:10), and for games that support the native resolution of my display it is quite breathtaking to behold. I've got 2 Dell 2405FPWs @ 24" 1920x1200 as well, which is what I used for gaming previously and it was very good as well for games that support the native resolution.
For games that are lower resolution it is nice to have on a bigger display though too even at the lower resolution, and the monitor scales things up better the more pixels there are in the hardware.
Of course what you say about higher pixels per inch being sharper is true at an equal distance from the eye. The larger screen gives a wider field of view though at equal distance and DPI which is nice.
As for a larger display being necessary, well.. I'd say that playing games isn't a necessity nor is even owning a computer necessary really. ;o) Buying a bigger or higher resolution display isn't about it needing to be necessary, it's about perceiving it to give benefits that are considered a worthy return for the money spent. When I bought my 24" displays the benefits it gave to gaming were fantastic. Necessary? Not at all. Enjoyable and worth every penny? You bet! Earlier this year when Dell put the 30" U3011 on sale it allowed me to stroke that piece of hardware off my perpetual wishlist, and I have to say that I wasn't prepared for the level of awesomeness that ensued with gaming. Massive eyegasms that must be experienced and can't be conveyed in words. ;o)
As for computer hardware, I had been loafing along with 10 year old hardware that was running on fumes and I decided it was time to build a new PC in February. Not just for gaming mind you but for a variety of other stuff that is CPU intensive, etc. as well. I put together a new system with an AMD FX8350, ASUS Sabertooth 990FX R2, Radeon HD7850, 32GB of Corsair Venegeance RAM @1866, and various other odds and ends for about $1000 total, well within my $1200 budget. The rest of the hardware I already owned, and the new monitor was a separate purchase before that (and not just for gaming either).
Having said all of that, I would agree with you that it is a bit expensive to spend a lot of money per year just for gaming alone and I wouldn't spend that kind of money on gaming either. In my case I needed new hardware and bought according to my computing needs (not just gaming). This machine will probably last a decade modulo a couple random upgrades here and there, so money well spent.
If someone has $1000 burning a hole in their pocket though and watches Dell's "days of deals" when they come around, and picks up a U3011 display, as long as your CPU and video hardware are capable of pushing 2560x1600 I guarantee you will foam at the mouth and start twitching from how awesome the gaming experience is. :) But yeah, kind of crazy to spend the money JUST for gaming. Fortunately I have other reasons for it so the gaming just bonused out from those. :)
Take care.
Nirth: I for one wants a 30" screen with 2560x1600 (100 PPI) but those are expensive and I would require a bad ass GPU as well. My 23" with 1920x1080 is "just fine" though, people draw the line differently. Also, 1280x1024 is a 4:3 format. That format sucks from both an aestetic and technical point of view. :P
Resolution is the last thing I lower in games to get increased performance too, it's such a down hill going from your native to something less when you're used to it.
I can say categorically that whenever you do end up able to indulge yourself in a 30" display, it's going to blow your mind. I am still in a daze and I've had it now for 4 months. :)
1280x1024 is 5:4 which is indeed one of the strange ones. I always opt to use 1280x960 (4:3) if I have to use 1280. I keep resolution as the last thing to lower as well.