"does the pixel at (0, 0) show the data from (−0.5, −0.5) to (0.5, 0.5), or from (0, 0) to (1, 1)?" - that's an interesting question, but it is independent of tile grids.
You're right that the "ideal datasets" of this world aren't discrete and pixelised, however, real-world pixel-based datasets exist, and real-world pixel-based screens exist, and you want to be able to explore large pixel-based datasets on pixel-based screens, which can be done in a really nice and crisp way if you just put one data pixel on each screen pixel. Suppose that these datasets live in some common cartesian coordinate system but have different extents. The issue of storing the tile ranges that cover each dataset is a real proposition - it's not "lossy and incorrect" once you've accepted that pixels are what you have to deal with as input.
If I want a system where zooming out aggregates 2x2 tiles to a single tile, then I have to decide what happens if there's an odd number of tile rows on a zoom level: Should the number of tile rows on the coarser zoom level be half rounded down, or half rounded up? It seems natural to take the half rounded up, and then fill out the missing tile row with blank pixels. What are the tile row numbers on the coarser zoom level? Suppose we start with tiles on rows 2,3,4,5,6 - that's [2,6] as a closed interval or [2,7) as an open interval. On the next zoom level, the tiles are aggregated so that rows 2 and 3 end up on row 1, row 4 and 5 end up on row 2, and row 6 ends up on row 3. This means we have tiles on rows [1,3] or [1,4). In general, if you have tiles on rows [a,b], then the next zoom level will have tiles on rows [floor(a/2), floor(b/2)]. Expressed with half-open intervals instead, if you have tiles on rows [a,b), then the next zoom level will have tiles on rows [floor(a/2), floor((b+1)/2)). I don't like the +1 in the formula for half-open, so I'll take the closed interval formulation any day.