Steve Foley writes:

Please explain how increased power can increase lift without first producing
increased velocity.

I never claimed that power alone would produce increased lift. However, an
increase in power will normally produce an increase in speed, all else being
equal, and so an increase in lift will result.

If the nose is pointed down (going downhill) , and you increase power, you
WILL descend faster.

Your increased speed will produce more lift, which will tend to raise the
aircraft, slowing the rate of descent and potentially leveling the aircraft or
producing a climb. For any given setting of AOA and thrust, the aircraft will
tend to converge on a specific density altitude. If AOA is held and thrust is
increased, the aircraft will converge on a higher final altitude than it would
if thrust were not increased. Increasing power (and thus thrust) at constant
AOA will produce a positive change in climb rate, because it increases the
speed of air flowing over the wings.