What role does the Holy Ghost fill that the Father can't? In trinitarian doctrine, what role does the Holy Ghost fill that the Father can't / doesn't?
My understanding is, that according to trinitarians:

The Father is a spirit
The Holy Ghost is a spirit
The Father can indwell someone
The Holy Ghost can indwell someone
The Father bears witness of the Son
The Holy Ghost bears witness of the Son

The Son, after his incarnation, has a body so it seems clear that he has a distinct role, and those point don't directly apply to him. But it's unclear what makes the Holy Ghost different enough in trinitarian doctrine to make him a distinct person / have a distinct role.
Regarding the mission (missio = a sending) of the the Divine Persons,

The Father cannot be sent:Summa Theologica I q. 43 a. 4 co.:The very idea of mission means procession from another, and in God it means procession according to origin, as above expounded (a. 1). Hence, as the Father is not from another, in no way is it fitting for Him to be sent; but this can only belong to the Son and to the Holy Ghost, to Whom it belongs to be from another.missio in sui ratione importat processionem ab alio; et in divinis, secundum originem, ut supra dictum est. Unde, cum pater non sit ab alio, nullo modo convenit sibi mitti; sed solum filio et spiritui sancto, quibus convenit esse ab alio. 

The Son is sent invisibly:Summa Theologica I q. 43 a. 5 co.:The whole Trinity dwells in the mind by sanctifying grace, according to Jn. 14:23: &quot;We will come to him, and will make Our abode with him.&quot; But that a divine person be sent to anyone by invisible grace signifies both that this person dwells in a new way within him and that He has His origin from another. Hence, since both to the Son and to the Holy Ghost it belongs to dwell in the soul by grace, and to be from another, it therefore belongs to both of them to be invisibly sent. As to the Father, though He dwells in us by grace, still it does not belong to Him to be from another, and consequently He is not sent.per gratiam gratum facientem tota Trinitas inhabitat mentem, secundum illud Ioan. XIV, ad eum veniemus, et mansionem apud eum faciemus. Mitti autem personam divinam ad aliquem per invisibilem gratiam, significat novum modum inhabitandi illius personae, et originem eius ab alia. Unde, cum tam filio quam spiritui sancto conveniat et inhabitare per gratiam et ab alio esse, utrique convenit invisibiliter mitti. Patri autem licet conveniat inhabitare per gratiam, non tamen sibi convenit ab alio esse; et per consequens nec mitti. 

Holy Ghost is sent visibly:Summa Theologica I q. 43 a. 7 co.:God provides for all things according to the nature of each thing. Now the nature of man requires that he be led to the invisible by visible things, as explained above (q. 12 a. 12). Wherefore the invisible things of God must be made manifest to man by the things that are visible. As God, therefore, in a certain way has demonstrated Himself and His eternal processions to men by visible creatures, according to certain signs; so was it fitting that the invisible missions also of the divine persons should be made manifest by some visible creatures.This mode of manifestation applies in different ways to the Son and to the Holy Ghost. For it belongs to the Holy Ghost, Who proceeds as Love, to be the gift of sanctification; to the Son as the principle of the Holy Ghost, it belongs to the author of this sanctification. Thus the Son has been sent visibly as the author of sanctification; the Holy Ghost as the sign of sanctification.Deus providet omnibus secundum uniuscuiusque modum. Est autem modus connaturalis hominis, ut per visibilia ad invisibilia manuducatur, ut ex supra dictis patet, et ideo invisibilia Dei oportuit homini per visibilia manifestari. Sicut igitur seipsum Deus, et processiones aeternas personarum, per creaturas visibiles, secundum aliqua indicia, hominibus quodammodo demonstravit; ita conveniens fuit ut etiam invisibiles missiones divinarum personarum secundum aliquas visibiles creaturas manifestarentur. Aliter tamen filius et spiritus sanctus. Nam spiritui sancto, inquantum procedit ut amor, competit esse sanctificationis donum, filio autem, inquantum est spiritus sancti principium, competit esse sanctificationis huius auctorem. Et ideo filius visibiliter missus est tanquam sanctificationis auctor, sed spiritus sanctus tanquam sanctificationis indicium.