There have been carbed road bikes with the fuel tank under the engine. Various late 1980s 125cc 2 stroke Gileras for example, where in some cases it released space to allow a storage area for a crash helmet where the fuel would normally be.
Problem generally attributed to this design is a more noticeable difference to hanling from the centre of gravity moving as fuel is used.
Many carbed bikes did use a fuel pump, allowing the engine to lean forward, with downdraft carbs to provide a straight path to the head, while avoiding needing the fuel tank to sit above the high mounted carbs. Yamaha FZ750 was possibly the first designed like this in the mid 1980s.
On a more modern injected bike with a large 4 stroke engine I suspect an under engine fuel tank would provide packaging problems for the exhaust (trying to wrap a plastic tank safely around a hot exhaust and catalytic converter). While the exhaust could be run over the engine to avoid this, this would result in more complicated exhaust routing, and interferring with the fairly critical airbox.