I've got a low end LED one that I use for presentations with a motorcycle group I ride with. It's OK, but you get what you pay for, and I didn't pay much.
I also have an older projector that uses a 100W tracklite halogen bulb. I bought it at a liquidation place about 15 years ago on a lark thinking it would be a total piece of junk especially considering I think it was $35 at the time. The picture quality is VHS-ish and certainly not super bright (but brighter than the photo makes it look) but...it works. Still works all these years later. Brought it camping a few times for my son's birthday parties and set it up just using a sheet hanging on a rope.
Also used it in the kids bedrooms just projecting on the wall when they had sleepovers and such.
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Neither this old cheap POS nor the newer cheap POS I have would work at all on our pool deck I wouldn't think given typical city light pollution.
Honestly, if it were me and I had a sheltered place to put it.....35-60" LCTV's can be had for cheap now and will perform way better than any projector in challenging lighting conditions.
It's been my experience if you really want to do the 250" outdoor bigscreen thing...you're going to be into a seriously big dollar projector.