Jerusalem was called the Holy City simply because of one inhabitant; it was the dwelling-place of God, and His presence there made it what no other city of the earth was, the Holy City.
'In Salem also is His tabernacle, and His dwelling, place in Zion,' Psalm lxxvi.

2.
'Blessed be the Lord out of Zion, which dwelleth at Jerusalem,' Psalm cxxxv.

21.
So wrote the Psalmist, and he was right.

God had chosen Jerusalem as His home on earth, His abiding-place, His dwelling; and so long as _He_ remained there, Jerusalem and all its surroundings was holy.

The mountain on which it stood was the Holy Mountain; the city itself was the Holy City; the courts of the temple were the Holy Place, the temple itself was the Most Holy Place, whilst the inner sanctuary, in which God's glory appeared, was the Holy of Holies.
But at the time of the siege of Jerusalem, God was leaving the city, it was no longer to be His dwelling-place, and consequently it was no longer to be called the Holy City.