OP’s issue is NOT the position on the hoods but reaching the levers on the drops. Rolling the bars forward will bring the levers CLOSER to him at a more neutral angle.
It will actually move the levers further away. It will only help him reach the levers if he puts his hand in a different place on the drops, which could be achieved without moving the bars at all. The relationship between the levers and the handlebars is determined by where the brifters are positioned on the bars not the angle of the bars themselves.
OP also says that the angle of the hoods tips his wrist upwards so having them set closer to level might offset this. Flipping the stem and therefore lowering the bar will increase the angle of his wrists.
Totally agree, having the hoods level with certainly help. Flipping the stem won’t have any impact on the angle of the wrists, unless you change something else as well.Moving the levers on the bars will achieve positioning of the wrists on the hoods and reach to the lever on the drops.
Even on a road bike, if your seat is too high, you don’t generally solve it by installing longer cranks.