As a follow up, My lanai is just the original screened type. I have a great golf course view and it faces east with a full roof and is 17' x 50', so I wouldn't enclose it. It sits 2 feet above ground level, so mud is not a problem. It is level with the house floor, but pitched away, and I had drains installed in the bottom aluminum rail so it can be power washed. Unless there is and wind driven rain, it only gets wet less than 2 feet from the edge.
Now, if I had a 6x8 birdcage facing west at ground level, I would have done something to improve that.

It certainly is individual preference. We would never enclose our lanai.
We enjoy our east facing lanai in the evening when shady and our west facing front porch in the morning shade.
With that being said, rain does blow into the lanai on occasion getting our outdoor furniture wet. We chose the furniture accordingly and it’s not a problem for us.
Most months we enjoy sitting outside in either location… Exception being when very humid which is mainly summer. Other months we find it pleasant.
Good luck with your search.

If you enclose...
 

We enclosed our Lanai. Now we have an "additional room" that can be used all year round. Put in a small ac and you're all set. :)

We own Sarasota model overlooking water with eastern exposure. Lanai 1,000sf and we love it screened and would not enclose. We have large screen TV and built in fireplace. I enjoy sitting out there with the fans on. I have indoor outdoor rugs and sunbrella furniture which dries quickly if it even gets wet which only happens in the birdcaged whirlpool area. We do have one retractable screen between lanai and birdcage which blocks any wind and rain. We can sit at table in pouring rain. I love the screen and recommend. I really never wanted another room as we have plenty of room in our home. Outdoor space priceless.

I think it largely depends on where you're moving from. We moved from near Buffalo New York so we completely enjoy our screened linai I almost everyday of the year If only to sit out in the morning on those cold days. Even the cold days are much warmer than anything we're used to in the off-season!

We wouldn't think of enclosing our Lanai.

We too move from up north, between Rochester and Buffalo. We have our screened and the floor tiled. We spend by far and large the most time on our Lanai every single day. Rain or shine cold or hot that's where we'll be. And we absolutely love it.

I know the feeling, having moved from between Rochester and Syracuse

Curious about drains in your railing. I would love this. Never saw them before. Where can I see them? Thank you.

Glassed in Lanai's can be added. We glassed ours in seven years after we built. I don't think you need make that decision now. Find a house that you like, where you like it, at a price you are comfortable with and buy it. But don't wait very long. They are selling quickly.

I think it fair to mention that all screened rooms here get very dirty, very quickly. VERY dirty.

It's personal preference. We bought a brand-new Sun Kist model and used the screened lanai that first fall/winter/spring. BUT then summer came (it faces the east) it was too hot....never used it. Then it got really dirty with the weather. about the 5th year we enclosed it...made it into an actual room. The floor is raised to meet the rest of the house. We had the slider moved to one end so there are glass sliding windows (with screens) that have a knee-height wall. Our house was small enough that more heat and air wasn't needed. I don't know that it has increased the taxes on our house but then again it's a small house. We've used it as a "sun-room". Had a couch out there and a chair or two. We would do this again.

****edit. Before the remodel it was essentially a small screen porch. It's set higher as the lot slopes so no mud ever splashed in. It was just dirty from dust etc.

There's a guy who installs them. They are about 1 1/2" high, 5-6" wide, and have a "swinging" door with perforations for water drainage that can open for larger debris. He cuts a hole in the base railing of the aluminum and these slide right in. I think they cost about $40 each, but you'll need multiple drains depending on the size of you lanai. I'm out of town now but will PM you if I can find out who it was and with some pix. If you don't hear in a week, send me a reminder

As you can see, opinions vary greatly. We have a closed in Lanai with sliders on 3 sides and it is perfect for us. There is plenty of air flow when it is cool enough or we can air condition it so it is usable year around. My sisters screened in birdcage requires a lot more cleaning and furniture wears out quickly just as it would if it was outside patio furniture.

We were in a rush to move here (our house in Va sold much faster than expected) and the lanai had a western exposure. Way to hot to stay there during the afternoon summer days.

We converted the lanai to a Florida room with double pain low e glass and so much better and gives us a lot more room.

Still debating putting in a split A/C.

Lanai
 

I love my screened in lanai. I love the fresh air. If you close it in, it is not a lanai anymore. It is just another room.