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

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I got a Freestyle 2 which I like I can move it in between rooms and can take it camping too
I bought a Freestyle 2nd Gen off eBay for a couple hundred bucks a couple months ago and haven’t regretted it even though this sub says not to touch it with a ten foot pole. I don’t want a projector to replace my TV, I wanted something inexpensive to take on trips or visiting family or to set up wherever, whenever. It fit the bill perfectly.
There's already a projector with these features built in, and one of the few I actually recommend.... go with the Samsung freestyle 2nd gen with gaming hub. The auto focus/frame, combined with built in apps out the box? Makes a lot of this incredibly easy. I have one, its been a beast.
They are not bad but dimmer than you expecting, maybe 300 ansi lumens max in usable picture modes, and contrast is low, 500:1 and below
The freestyle 2 is like 300lm, I have one it sucks outdoors even at night. Go with a 1080p laser projector from benq, viewsonic, or optoma. You should be able to get a 4000+ lumen one for around $1000.
I got the Samsung freestyle second gen on a cracking deal a couple of years ago and it's all I use. Works as a smart projector so you don't need to hook anything up just connect to WiFi. I also have a Chromecast connected to it for anything it can't in the dodgy streams. You can use the browser on it to get to them but that's a bit of pain without a keyboard and ad blockers. Works on a wall when watching by myself or when we are watching together we use the screen. It auto focuses and auto levels so it's great highly recommended and the speaker on it is quality for our small room.

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Great value with huge lens shift, but outdated lamp tech and poor HDR.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
JMGO - N1S 4K

Ranked #1
Anker - Mars 3 Air