Is an increase in induced drag always caused by an increase in the angle of attack?
At 2:58, the video says "To replace the lift lost by the increased upwash and downwash, the wing must be flown at a higher-angle-of-attack than would otherwise be necessary."

This statement is true no matter what angle-of-attack the wing is "feeling" and no matter how much lift it is making, as long as the amount of lift it is making is not zero.

For example, let's say at a given instant in time, with a given angle-of-attack and airspeed, when the instantaneous flight path is horizontal, the wing would be making Lift equal to Weight if it weren't for the upwash and downwash effect, but because of the upwash and downwash effect, the wing is only making a force equal to 90% of Weight. 

Forces are not in balance, and the flight path is curving downward.  In the next instant in time the flight time will no longer be horizontal.  Yet despite the fact that we aren't actually replacing the "missing" lift, it is still true that that "To replace the lift lost by the increased upwash and downwash, the wing must be flown at a higher-angle-of-attack than would otherwise be necessary."  If we were happy to settle for a lift force equal to 90% of weight at that instant in time, and the upwash and downwash effect were not there, we could fly at an even lower angle-of-attack.

More practically, we normally don't want to allow the flight path to curve (accelerate) downward, so we do keep lift close to weight, or very close to it, except during the brief transition (pushover) from a climb to level flight or from level flight to a descent.  The reduction of lift that happens in a steady-state descent (or climb), compared to in level flight, is very small for moderate climb or descent angles.  For more, see a What produces thrust along the line of flight in a glider? and Does lift equal weight in a climb? .

Only if the wing is mounted inside a wind tunnel, where the connection between Lift and Weight is completely severed, does it really make any sense to ask "What happens if we decide not to replace the missing lift?".  In that case, indeed, the upwash/downwash effect does not force the wing to be flown at a higher angle-of-attack than would otherwise be necessary.