I think we're on the same page with the screen enclosure. Good visibility. Good circulation. Pickets would look nicer, but those oak leaves are tiny and are going to get through the spaces between the pickets. You could always add pickets over the screen, for aesthetics.
Five posts in concrete (one extra for the door) with some 2x4s top and bottom to support the screen. 2x4 door frame. Use fiberglass screen, or 1/4" hardware wire if you need some animal proofing and/or want it to last forever. Two of the posts would be an inch or two taller, to give the roof some slope. Then metal corrigated roofing on top. Cheap, comes in colors, will last forever, and is relatively easy to work with. Some places will even cut it to length for you, if you order it that way (Lowes did for me). You'll need a bit of support in the middle of the roof, which can be a few 2x4 "rafters." Use screws and cut the screen such so that at least one wall could be removed, so that you could disassemble it if you ever needed more access than the door allows. Bobs-yer-uncle. Probably $200-300 in material and an afternoon or two of labor...
If you wanted to get more elaborate, you could frame in some shelves with some painted plywood sides and doors to accommodate storage: toys, pool hardware, muriatic acid (chlorine should be stored somewhere cooler), and what-not. A couple of hooks for pool brush and leaf rake, etc.
Don't make it too big or too nice or the significant other will start to get ideas about how to reassess which rooms are "just for you" and which are for the both of you (meaning hers!)...
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