Correct, the non-Catholic minister would not be officiating on behalf of the Church, only on behalf of the civil law. It's also correct that for the conditions of Canon 1098 to be fulfilled, apart from danger of death, it must be prudently foreseen that the unavailability of the pastor, ordinary, or delegated priest will last for one month. And it's also true that any other Catholic priest should be called under these conditions to perform the marriage. It will be valid, though, without him, as Fr. Augustine says, "“Lastly, calling a priest does not affect the validity of the marriage, which therefore may be contracted validly in the presence of only two witnesses.” (Augustine, Charles P., O.S.B., D.D. A Commentary on the New Code of Canon Law. 3 rd Ed. Vol. V, Book III, St. Louis, MO: B. Herder Book Co., 1920. C.1098, p.296).
Among other things, this priest would serve as one of the two required ordinary witnesses, but he would not be an official witness like the pastor, ordinary, or delegated priest.
"But since such a priest is not a testi qualificatus seu authorizabilis, “neque ad valorem neque ut videtur ad liceitatem matrimonii necesse est ut requirat excipiatque consensum.”
-- De Reeper, John, M.H. “The History and Application of Canon 1098.” The Jurist, Vol.14. Washington, D.C.: The Catholic University of America Press, 1954. p.171.
The only requirement of the two witnesses is that they have reached the age of reason and be able to act as witnesses. They are not required to ask anything or receive the mutual consent like a priest. They only need to be present in person and together, knowing that a marriage is being contracted between those two persons. Even if the parties unexpectedly exchange consent in the presence of the two witnesses, the marriage is valid under the conditions of c.1098.
“The second essential condition requires that two witnesses should be present. The canon uses the expression ‘testes’ in the plural, so one witness is not sufficient. The witness need ask nothing and need not receive the mutual consent, as must the priest when assisting. It suffices that the consent should be given in their presence. So they must be present: personally (not by telephone), together, ‘moraliter’ i.e. the must know that a marriage is being contracted between those two persons. In order to be valid witnesses, i.e. ad validitatem, the only requirements are that they should have reached the age of reason and be able to act as witnesses. So male or female, puberes or impuberes, Catholic or non-catholic, voluntary or forced, warned beforehand or totally unexpected, -so long as the ‘actu’ witness the contracting of the marriage.”
-- De Reeper, John, M.H. “The History and Application of Canon 1098.” The Jurist, Vol.14. Washington, D.C.: The Catholic University of America Press, 1954. pp.170-171
“Coram solis testibus. Since the Codex uses the plural, it goes without saying that one witness will not suffice. They need, however, ask or do nothing even if the parties should unexpectedly exchange consent in their presence, that would be sufficient to satisfy the demands of the law; it suffices, therefore, if they are merely present.”
-- De Reeper, John, M.H. “The History and Application of Canon 1098.” The Jurist, Vol.14. Washington, D.C.: The Catholic University of America Press, 1954. pp.175
I'm not aware of anywhere that it's possible for Catholics to marry in the presence of the ordinary, pastor, or priest delegated by either of those, as prescribed by c.1094. Such conditions have existed for years now, and it may be prudently foreseen that these conditions will last at least another month. It seems reasonable to conclude that a majority of Catholics are currently under the conditions described in c.1098.