
Valerion
VisionMaster Pro 2
Great picture, good gaming, but pronounced RBE and no lens shift.

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Good news, your expectations are super reasonable. Do you want one with a built-in OS like Android or are you happy just throwing a Fire Stick or equivalent in for streaming movies? XGIMI has refurbished models on their website around your price point that'll likely exceed what you're looking for, and you can very likely find a used BenQ or an older-model Valerion around there too.
You should be able to get the XGIMI Horizon refurb on their website around that. It's what I'd be using if I hadn't gotten a great deal on my TK710 this summer (Amazon Prime deals can be a heck of a thing if you get lucky)
We currently have this one https://www.bol.com/be/nl/p/xgimi-horizon-smart-projector-beamer-harman-kardon-2-x-8w-luidsprekers-chromecast-1500-iso-lumen-dolby-audio-bluetooth-dts-h/9300000044603293/?Referrer=ADVNLGOO002013-S--9300000044603293-PMAX-C-22290424073&gad_source=1&gad_campaignid=22294182655&gbraid=0AAAAAD5OnmPlILNzP11NiqlX2MtiWrlqW&gclid=Cj0KCQiAi9rJBhCYARIsALyPDts1akWALTyWymsIadPHF43_jL_Kkimu6EWjVmg1IzPJ8DNfdxi53CIaAuGMEALw_wcB But we have also had the Halo+ (is think) before that. We are verry happy, this thing runs Every day for hours on end (im partial insomniac, so spend most time watching shows and movies on Plex). This thing is great
Xgimi horizon pro max. Lens shift works beautifully. Optical zoom capability, Software is snappy. Projected image is crisp without light spills
Think of not only gaming, HDR, lumens shmumens but maintanance as well. if you buy a projector as per your example - prepare to replace the lamp every 1-2 years depending on screen time. I'd steer you toward LCD or laser projectors. I personally have XGIMI Horizon Pro 4K Projector, 2200 ANSI Lumens, it has gaming mode and ultra fast gaming modes. with these projectors the screentime untill issues is 30k, lamp 15k. At the same time I never owned a laser projector but I believe these are best for short throw distances. P.S. the XGIMI recently released a new projector and I heard people praising it. as for myself - I have this projector for 2 years and 0 issues the only inconvinience I got is that I can't just install steam remote play onto the projector, I believe that's due to it having 2 separate boards one for android and second for projection which is responsible for gaming at low latency quoted 18ms so HDMI gaming only, the other possible inconvinience would be netflix app not available but I love my jellyfin, cloudstream apps and to read r/Piracy they got some fun articles there. [https://a.co/d/hxJRcfE](https://a.co/d/hxJRcfE) \- here's the new version, it's way under your budget and I guess covers all of the things.
Depends on the size screen youre tryig to have. I disagree with other commenter on projectors. they arent hard to set up, they are easier to move, and if you want a large screen way outpace an oled, because pricing gets high for big OLED tvs and the bigger they are, the more stationary they becime. A decent 4k projector starts around $1000 you can get a 65inch oled for that price. If youre going to be using a projector at 80inch screensize or smaller, get an oled in your price range, the image quality of oled is unparalleled, But if yiu want 80inches or bigger projectors are awesome, and most play 3d too. However these days you can get a 77inch oled for $1500, so not a huge difference,and high end projectors will cost at least 2x that. I have a 42inch c2 oled in my bedroom, a xgimi 1080p projector casting about 75-80inchs in there as well that only cost $200 and I use both alot depending on what im watching and lighting conditions. In my family room I have a 42 inch cheap 4k tv, and the xgimi horizon s pro (about $1200 when i bought it) 2024 model 4k projector casting about 120inch screen. I never even turn on the 42inch in the family room, the projector is just so captivating because the size, and while image quality is less than that of an OLED, its still beautiful and 3d movies that size are fun to watch as an added bonus. Gaming on a huge screen is also just a lot of fun. Bright light and basically all projectors become bad, so if you have no control over lighting and watch a lot during the day, a projector is no good. Like everything, there are positives and negatives to both.
I would look up the xgimi h2. You can pick that up second hand or pretty much new for less or you could pick up a second hand horizon pro 4k. In Spain I’ve seen them used but in good condition for around €590. We’ve had our h2 for almost 7 years and it’s still going well as our main tv. We have a few dead pixels so considering upgrading later this year to a 4k model.

Valerion
VisionMaster Pro 2
Great picture, good gaming, but pronounced RBE and no lens shift.

Epson
Home Cinema 5050UB (V11H930020)
Best value, flexible lens shift, but dated tech, poor HDR.
Epson
Pro Cinema LS12000 4K PRO-UHD Laser Projector
Great 4K/120Hz gaming, no RBE, but needs calibration.

JVC
DLA-RS1200 / DLA-NZ500
Unmatched black levels for movies, poor for gaming.

BenQ
HT2060
Great 1080p gaming, deep blacks, but low brightness.

Ranked #1
Valerion - VisionMaster Pro 2
Ranked #1
Epson - Pro Cinema LS12000 4K PRO-UHD Laser Projector

Ranked #1
Hisense - Laser TV PX3-PRO

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

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

Ranked #1
JMGO - PicoFlix