Our Lady of Grace is under the omophor of the Latin Archbishop of NY because, when it still had a temple of its own, it was situated in Manhattan (Broome St originally, then Stanton St, and - after the repose of Papa Ciro, of blessed memory - when it was reduced to the status of a Shrine, Mott St), all geographically situated in the Archdiocese. It currently has no temple and its location (mailing address, really) is Archdeacon John's home in Staten Island - which is in the Latin Diocese of Brooklyn.
As the hierarchs of the Brooklyn Diocese have a long history of benevolence toward Eastern and Oriental Catholics (witness Brooklyn's provision of a home for the Armenian Eparchy after NY evicted it from its Cathedral - St Anne's), it seems most unfortunate that the Italo-Greek-Albanians have apparently made no effort to translate to the omophor of Brooklyn. I suspect, though, that they may be precluded from doing so, by reason of some obscure reasoning that NY was where they last existed as an true ecclesial entity.
My guest about why Our Lady of Grace being under the Roman Bishop of New York, is that he can afford to take care of the mission. Especially since it only meets once a month
The Cardinal can hardly be said to be taking care of the mission. It meets where it can, most recently a UGCC parish - through the kindness of the Eparchy of Stamford; prior to that, and still occasionally, at the Byzantine chapel of the (Graymoor) Friars of the Antonement. To the best of my knowledge, the Society (its current status) of OLG receives neither any funds nor support of any kind from the Archdiocese of NY. Each successive Bishop of the Brooklyn Diocese, otoh, has ordinarily attended the Society's annual banquet/awards ceremony.
Regarding the prospect for them to be transferred to the omophor of a Byzantine Eparchy - unlikely. Our Lady of Wisdom, albeit serving an Italo-Greek community, was erected by the Van Nuys Eparchy through the kindness of Vladyka George, of blessed memory, not transferred to his care.
Other instances in which parishes of the "orphaned" Churches have come under the spiritual omophor of non-Latin hierarchs all have some unique aspects.
Father Archimandrite Alexei (Smith), pastor of St Andrew's Russian GC Church in El Segundo, happens to have been educated by the Melkites, serves also as Administrator of St Paul's Melkite Mission - with which the Russian parish shares its temple, and was ordained by the Melkite Eparch to the service of the Russians. Although the Russian parish remains under the canonical jurisdiction of the Latins, Cardinal Mahony asked that the Melkite Eparchy accept the Russian parish under its spiritual omophor - a very practical consideration, given the dual status.
The Melkites informally (LA is a formal situation) afford episcopal care (ordinations, etc) to OL of Fatima Russian GC Church in San Francisco - but, again, one of the priests serving there is incardinated to the Melkites.
As regards St Michael's Russian GC Church in NYC, Father Economos Romanos (Russo) of the Melkite Eparchy is presently serving it, but it is not clear to me that our Eparchy has any standing - even informally - with regard to the parish.
The Romanian Eparchy of St George in Canton informally affords episcopal care to Ss Cyril & Methodius Russian GC Mission in Denver by invitation of the local Latin hierarch, who retains canonical responsibility for it.
In Australia, the Russian Centre is, likewise, informally entrusted to the Melkite Eparch, essentially because Father Archimandrite Lawrence (Cross) is incardinated to the Eparchy. A (bi-ritual, I believe) priest of the Melkite Eparchy there is also occasionally serving the Australian Italo-Greek-Albanian community, but the community is not canonically organized.
There's little precedent for any of these cross-jurisdictional mixes among the Eastern Churches, except that of Archbishop Joseph (Tawil), of blessed memory, who was invited to provide episcopal care to OL of Kazan Russian GC Chapel by Cardinal Cushing, also of blessed memory. The fleeting nature of such, however, is readily observable by the fact that when the Cardinal's successor elected to suppress the Chapel, I have no recollection that Archbishop Joseph was offered any input into the decision.
As to the likelihood of an Exarch being named for the Italo-Greek-Albanians in the US, neither the Eparchs of Lungro degli Italo-Albanese nor Piana degli Albanese have the authority to do so. Any such action would, of necessity, have to originate from the "Colonial Office" and would almost assuredly require: an indication that OLG had a viable community; that other such existed elsewhere in the US; and a recommendation for such from some hierarch on this side of the Atlantic.
An effort, a couple of years ago, to establish a presence for the Italo-Greeks in New Orleans was rejected by the Latin hierarch there, despite a potentially sizeable community (reportedly larger and more viable than that in the NYC area). Those not absorbed into Latin parishes there worship with the Ruthenians or the Melkites (or did, until the Melkite mission went dormant post-Katrina).
Unfortunately, the Italo-Greek-Albanians - despite the work of OLGS - are little organized on this side of the world. Enough such that a priest of one of the Italian eparchies, allowed by his hierarch to travel to the US because of critical medical needs on the part of the part of his conjoined twin daughters and now here permanently because of their ongoing medical issues, is serving (last I knew) in a Ruthenian Eparchy - why? Likely because there is nowhere that a community exists which could support him and his family in an established parish environment.