Why can nobody agree which river is longest?

Because where a river begins is a decision rather than a fact. A river is fed by tributaries, each of which has tributaries, and choosing which headwater counts as the source determines the length - so the Nile-Amazon question turns on whether you accept expeditions that pushed the Amazon's source further into the Peruvian Andes, which produce figures beating the Nile.

The mouth is equally arbitrary at that scale. The Amazon's estuary is enormous and tidal, and where the river stops and the ocean starts is a judgement; measure to the far end of the Para channel and you gain hundreds of kilometres. Meandering makes it worse, because a river's length depends on the resolution you measure it at - the same coastline paradox that makes national coastline figures unreliable.

So the honest answer is that both claims are defensible and the ranking depends on choices made before any measuring starts. The Nile is the traditional answer, the Amazon has the better recent expedition data, and neither camp is being dishonest. By volume there is no argument at all: the Amazon discharges more water than the next seven rivers combined.