I think it comes down to the individual's view of whether it is Biblically okay for a layperson to pray directly to God, or whether they instead believe there must be an "official" intercessor, such as a living member of a given priesthood or clergy, or a designated saint. Further, it depends on how active they are in prayer and seeking the Holy Spirit in their daily lives.
If a Christian individual follows the exhortations in James to have an active prayer life, and they pray daily without just moving their lips or repeating words they have memorized, but from their hearts with trust and intimacy, then they are more likely to accept any Christian prayers, Jewish prayers -- any that are, as you say, well intended. (I'm making an exception here for that religion that plays "gotcha" and will only pray for the other person to convert to their belief -- or else!)
That makes logical sense.
But I don't know if it's germane to this particular discussion, since I gather there's nobody here who doesn't think it's "Biblically okay for a layperson to pray directly to God."
Did I miss soemthing? (It's always possible...)
So I think this particular discussion probably revolves around whether it was a designated liturgical blessing, or on the other hand, an ordinary blessing such as anyone could do. As I said elsewhere, I can't give the nuptial blessing given by the priest or deacon at the Sacrament of Matrimony, but I can certainly say "God bless you 10,000 times!" or any other blessing I feel needs to be said!
BTW, my Confirmation Name, Benedicta, is by happy coincidence a pun in Latin: it means both "Blessed" and "Well spoken" ...Bene-dicta. :o) .. a hopeful thought!