What you're referring to is tucking, an act that's most easily done with drop bars - the curved handlebars with multiple hand positions you see on road bikes. The holy grail for some hardcore roadies (road cyclists) is the flat back, where the rider is tucked all the way forward, their hands on the lower part of the drop bars - the "drops".
This is done so that the cyclist will present less surface area to the wind, and therefore there will be less drag. The "flat back" is most easily achieved when the saddle (the seat) is higher than the handlebars.
To my knowledge, placing more weight on the front wheel is mostly incidental here.
Hybrid bikes almost always come with flat handlebars, so while it's technically possible to tuck to the degree a road cyclist does, it'd be intensely uncomfortable to maintain. The distance between the seat and the handlebars is a factor here, but less of one than saddle height. (You can look up questions on bike-fit for more information on these variables, since fit is a bit of a black art and there's a lot of information available on the subject.)