The principal behind the deep sump oil pan is it keeps the oil pool further away from the spinning crankshaft. At high RPM, a crank can actually suck up oil from the sump and spin it up into the area around the reciprocating assembly. All this oil in the air is then impacted by the spinning components. The impact is similar to when you slap your hand into water, it slows things down and costs horsepower. In a race engine, this loss can be 15-20 horsepower. For a street car that does not consistently run at high rpm, this impact is considerably less and may not even be noticeable.
For a road going car, an original style pan would not be an impact to driveability. There are pans offered by manufacturers that are not deeper than stock, but are wider to hold additional oil capacity without giving up ground clearance. Mancini Racing is a good source to review oil pan options.
