I found five books in my possession that deal with Samuel Rutherford, either entirely, or in parts. Despite spending time searching for any reference Rutherford made to marriage, I could not get even one. However, I gleaned a few points that might have a bearing on a claimed stance of disagreeing with Christian pastors presiding over marriage ceremonies. This is linked to his views on the powers of the civil magistrate.
Here is where the Westminster Confession makes an essential first point, namely, that marriage is not one of the sacraments. As has been pointed out, Rutherford spent a lot of time in London taking part in the Assembly. Before he traveled from Scotland in 1643 for this, he had lost two of his four children in death. He left the Assembly on 9th November, 1647, which ended a few months later. Then we read this:
"I had but two children, and both are dead since I came hither."
Rutherford, Letter No. 310, as recorded in The Life and Works of
Samuel Rutherford, p.34, James Clarke, M.A., 1986
Rutherford took a prominent part in the Assembly that drew up the Westminster Confession. A Catechism exists, in Rutherford's handwriting, very much resembling the Westminster one. From this, we can be sure that he agreed that there were only two sacraments: baptism, and the Lord's Supper (XXVII.4). Therefore, it is not surprising that there seems to be very little in Rutherford's writings about marriage. However, that is not my area of expertise, so a scholar might be able to show a collection of such writings.
In Section XXIV - Of Marriage and Divorce - there are some interesting points about marriage (1) That marriage as a divine institution is clearly taught in scripture. (2) That where civil law recognizes a minister of the Church as an agent of society with civil authority to sanction the marriage of unbelievers, it may in some cases be his duty to so act, that a greater evil may be averted. But it would be the duty of a minister, in such a case, to inquire as to the religious profession of both parties and to proceed only if both make no profession of the true religion. If one, and only one, does profess the true religion, however, it would be contrary to the Word of God to sanction such a marriage. For the scripture clearly states that a believer must marry "only in the Lord".
Those two points have been taken from The Westminster Confession of Faith for Study Classes, by G.I. Williamson, p182, Presbyterian and Reformed Publishing Co., 1964.
From this snippet of information, I suppose that Presbyterian clergymen (like Rutherford) would be expected by the civil authority to officiate at marriages, but that they would not marry certain couples. Combining that with not viewing marriage as a sacrament, that may have given rise to the idea you heard that "Samuel Rutherford and others did not believe that Christian pastors should preside over weddings – that instead, it was the civil magistrate's role to marry people." It seems more likely that only in exceptional circumstances, where the pastor's conscience did not permit him to officiate at what, in scripture, would be a unsanctionable marriage, would it be left to the civil magistrate to take over.
But given that I have not been able to find any of Rutherford's actual words on the matter, this question remains open. I simply offer, as a basis for my answer, Rutherford's agreement with the Westminster Confession regarding marriage.