
Valerion
VisionMaster Pro 2
Great picture/gaming, but RBE and no lens shift are dealbreakers.

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You don't really get 'better' projectors for less money, but you get better for the same amount of money. Based on your room size and request for as large of a screen as possible, I would look at the BenQ TK700STi. It would need to be a refurb, and still may not be within budget. It can fill the wall you have edge to edge without problem from 8.5' or so lens to screen (throw distance). Learn how to use this tool and what all the terms mean... [https://www.projectorcentral.com/BenQ-TK700STi-projection-calculator-pro.htm#calc](https://www.projectorcentral.com/BenQ-TK700STi-projection-calculator-pro.htm#calc) There aren't a ton of 'short throw' projectors on the market, but they do have their uses.
Look into BenQ TKI700STI, it's 4k with good brightness and shorter throw distance than regular projector. I got one and I have been very happy with it, ofc image is not as good as TV, but I appreciate the size over TV like quality. But the image quality is still pretty good! I find it usable for not so intense watching on daylight even during summer, but if I want to have a gaming session or watch movie on it then I prefer to completely blackout the whole room. I got 120" picture from around 3m throw distance.
I have the Benq 700 STI, 4K, 3200 lumen, practically no latency (if you want to play games), you can easily find it around 1000€ 😉
Check marketplace I just got a BenQ700STI 4k projector with 500 hours for $800.
I have a Benq TK700sti, which is a short throw, that I use as a monitor in my media room. I bought it explicitly for the low input lag and motion clarity, I like it a lot in that use case. It is good for watching movies too, but not top tier. The projector is an older model now that the tk710 is out, but it runs 240Hz at 1080p out of the box and a pixel-shifted 4K at 60Hz. It is DLP and I can see the rainbow effect on star-fields at 24Hz, but above 120Hz I never notice it. DLP has better motion clarity than most other projector techs because the rendering of the three different colors at different times give an effect similar to black frame insertion. This also allows the projector to display for 3D movies. My room is blacked out, but it's a bright enough projection that it can be used in low light as well, one will just go through lamps quicker. The tk710 is probably the better buy currently, but I've heard reports of light bleed on the earlier manufactured models. My projector is ceiling mounted on an 8' high ceiling and my viewing position is 8' from the 115" screen. Any closer and I have to move my head to see action at one end or the other of screen. My seating position is pretty much directly under the projector, and I don't obscure the projector beam at all when I stand up, so a short throw might be the way to go if your rig is big. I'd be wary of any software shifting of the projected image, as it may introduce input lag. Using a Keystone feature definitely does. I think a 2 or 3 projector setup for simulators would be amazing.
For a 60-inch screen with the BenQ GP500 (500i), the projector is designed to sit level with the bottom edge of your screen. Because it has a 100% offset, if you place the projector on a table, the image starts right at the lens height and projects upward. If your table is significantly lower than the screen, you will have to tilt the projector and use Auto Keystone, which can slightly reduce image quality and create a faint "glow" around the edges of the frame. Recommendation: For the best results, try to get the projector lens as close to the height of the screen's bottom edge as possible. If gaming is your priority, the BenQ TK700STi is a superior alternative under $1,600 CAD due to its much lower input lag.

Valerion
VisionMaster Pro 2
Great picture/gaming, but RBE and no lens shift are dealbreakers.

Epson
Home Cinema 5050UB (V11H930020)
Great value with huge lens shift, but outdated lamp tech and poor HDR.
Epson
Pro Cinema LS12000 4K PRO-UHD Laser Projector
Superb 4K/120Hz gaming and no RBE, but requires calibration.

BenQ
W1070
Budget king, durable with good image, but RBE can be distracting.

BenQ
HT2060
Excellent 1080p contrast and quiet gaming, but low brightness for big screens.

Ranked #1
Valerion - VisionMaster Pro 2

Ranked #1
Epson - Home Cinema 5050UB (V11H930020)

Ranked #1
Hisense - Laser TV PX3-PRO

Ranked #1
Anker - NEBULA X1 4K Triple Laser Projector (D2351)

Ranked #1
BenQ - GV50 Portable Projector

Ranked #1
Anker - Mars 3 Air