Yes, He did. The Tabernacle is, of course a direct prefigurement of Mary, and the Pentecost is when the First Church originally most closely personified in Mary is now blessed in the full assembly of her episcopacy. These are all types of the cardinal event, the Incarnation.
God also overshadowed the Jews in battle. That is the reference used by the Psalmist. But so does Mary call us to battle (Apoc. 12:7-17) -- we are, after all Church Militant. This, as well as the descent of the Holy Spirit at the Pentecost explains to the Catholic faithful that the Holy Spirit never left Mary, the Mother of the Church, the Spouse of the Holy Spirit.
Prior to Leo XIII’s Divinum Illud Munus, are you aware of any Father who called Panagia the “Spouse of the Holy Spirit”? I do know that +John of Damascus calls her the “spouse of the Father” and +Germanos of Constantinople (+733) called her “Theonymphos”, wedded to God while Blessed Augustine refers to her as “Spouse of God”, but these are virtually unique comments.
From what I can see, this “Spouse of the Holy Spirit” business has become very widespread among Roman Catholics over the past 100+ years and it seems to be nothing more than the same sort of religious “enthusiam” which has lead to the heretical “Co-Redemptrix” nonsense or seeing the Most Holy Theotokos in a damp stain on a bridge abutment.
I have to say, Alex, that this rationalizing of the Incarnation by a focus on legitimatizing it by a faux matrimony, is vaguely distasteful. Perhaps that’s why such a notion never gained any Patristic traction.