Uhhh...that's cheaper because it doesn't do what a roof does.
It doesn't insulate and it doesn't reflect the heat away from the house.
You essentially need an 'underroof' at that point, something on top of the ceiling that serves the purpose of the roof.
It is not cheaper than normal to build a house with a flat roof and then build a greenhouse on top of that roof, which is what you've actually suggested here.
And it's certainly not cheaper in any climate that would be described as 'tropical'.
Even with an older, fairly non-reflective roof, the heated up part is the 'outside' of the roof, whereas with a greenhouse, the heat would end up being on the 'underroof'. Even if you open the greenhouse totally to the outside air, it's still worse than a normal roof, which at least has an air buffer which it has to heat first. (And that itself can be vented.)
Such a greenhouse might make sense in cold climates where cooling is never an issue, although freezing and snow would present interesting problems...your roof would not heat up as quickly to melt the snow, hence removing some of the 'greenhouse effect' you're counting on to heat the house, and resulting in a freezing 'greenhouse'. (Which is hopefully insulated, via the underroof, from the actual house, or all the hot air ends up in there.) OTOH, you could hook up 'defrosters' like in cars and melt the snow from the inside.
I'm not sure what collecting roof runoff has to do with anything. Most traditional roofs have gutters, and, hence, could easily collect runoff if people wanted.