> If the mortgage is covered by extortionate rents on your other properties then historically the empty unit is appreciating in value.

That appreciation is happening whether the unit is rented or empty.

The scenario the landlord sees for a given unit is (assuming zero mortgage costs just to make the math simpler [1]):

Empty unit: $0 in rent + $Y in value appreciation

Rented unit: $X in rent (X>0) + $Y in value appreciation (where Y is fixed between the two scenarios because both cover the same rental unit).

For the landlord, $X + $Y is greater than $0 + $Y.

The landlord's greatest total income is when all units are rented, as then all units are returning $X+$Y.

[1] including mortgage and tax costs adds two subtractions to the formulas above, which will result in a real net loss for an empty unit, as the mortgage and taxes must be paid, irrespective of the rented status of the unit.