I have been trying to identify the 'point of contention'. The Anglican position is, perforce, terse:
Sacraments ordained of Christ be not only badges or tokens [notae] of Christian men's profession, but rather they be certain [certa] sure witnesses and effectual signs of grace [efficacia signa gratiae] and God's good will towards us, by the which He doth work [operatur] invisibly in us, and doth not only quicken, but also strengthen and confirm, our faith in Him.
The Supper of the Lord is not only a sign [signum] of the love that Christians ought to have among themselves, one to another, but rather it is a sacrament of our redemption by Christ's death [sacramentum nostrae per mortem Christi redemptionis]: insomuch that to such as rightly, worthily, and with faith receive the same, the bread which we break is a partaking of the body of Christ [corporis Christi], and likewise the cup of blessing is a partaking of the blood of Christ [sanguinis Christi].
Wherefore the sacrifices of Masses, in the which it was commonly said that the priests did offer Christ [sacerdotem offerre Christum] for the quick and the dead to have remission of pain or guilt [in remissionem poenae aut culpae], were blasphemous fables and dangerous deceits.
The sticking point is thus bolded. Of course, many will reply, "that is not at all what we mean to say." But consider, nonetheless, the content of the following set of utterances, which are all too typical:
Thirdly; Mass is offered to obtain pardon of our sins. Two things are to be considered in sin (1) its guilt; (2) its punishment. Mass as it helps to the forgiveness of sin is propitiatory, in its power of cancelling punishment it is satisfactory. The Council of Trent teaches (Sess.xxii. ch. 2) that this "Sacrifice is truly propitiatory, and that forgiveness of sins and of enormous crimes is obtained by those who with a true heart and right faith, with fear and reverence, contrite and penitent, approach to God." The Mass then obtains the pardon of mortal and venial sins and of the temporal punishment due to sin.
The Mass as propitiatory appeases the anger and justice of God. "The Lord, being appeased by the offering of this Sacrifice, granting grace and the gift of repentance, wipes away crimes and even enormous sins." (Council of Trent, Sess. xxii. ch. 2.) A distinctive effect of this Sacrifice is that by it God is appeased, as a man forgives an offence on account of some homage which is paid him. For Mass does not forgive sins directly and immediately, like Baptism and Penance. Mass appeases the anger of God, and obtains from Him the grace of repentance. Man can, if he chooses, reject the grace and remain in sin; the free acceptance of this grace enables the creature to turn to God by Faith, Hope, Charity, and Sorrow, and thus to receive worthily those sacraments which of themselves forgive all his sins.
The propitiatory power of the Mass disarms God's justice; the impetratory power draws down His mercy. Indirectly Mass causes the conversion of sinners as a propitiatory Sacrifice appeasing God’s anger, leaving scope for His mercy; in so far as it is impetratory, it obtains the grace of repentance, which may be accepted or rejected. The propitiatory power is infallible as Christ’s work, that is, the Lord is in some ways appeased, though to what extent never can be known. This depends on the free-will of God and on the dispositions of the creature.
The power of the Mass to forgive sins is more clearly understood by selecting a particular case. Let us take a simple illustration. Suppose a mother has a Mass offered for each of her sons, John and James. John is leading a bad life; James is a practical Catholic and is free from mortal sin. What effect on John has the Mass said for him? It may be altogether barren of result, because John can reject, if he likes, "the grace and gift of repentance," which the Council of Trent speaks of. (Sess. xxii. ch. 2.) We are certain at least of this; first, that Mass necessarily and infallibly appeases to some extent the anger of God which John has provoked by his sins; secondly, that it obtains from God necessarily and infallibly grace which, though not always of itself sufficient at the moment to cause John’s conversion, goes some way towards it. Many Masses may be needed before John’s conversion is secured. If John does what in him lies he will get further grace to stir his heart to repentance and to seek reconciliation and pardon in the Tribunal of Penance. The Council of Trent, in the passage quoted above, must not be under stood to teach that Mass of itself forgives "enormous crimes." Mass does not forgive the sins of John. Mass wins for John, supposing he accepts and uses the grace offered, the additional grace to make a good confession, and thus to have his sins forgiven. Let us now turn to James, who is free from grave sin. What benefit does he receive from the Mass said for him? First, that Mass as the action of Christ, who is the chief Celebrant in every Mass, necessarily and infallibly satisfies for some of the temporal punishment due to past sins, the guilt of which has been forgiven; secondly, it obtains fresh graces for James, strengthening him against temptation or fall, enabling him to lead a holier life and to persevere in God s service.
By Mass also (Council of Trent, Sess. xxii. ch. i) we obtain forgiveness of daily small faults through those actual graces which stir us to sorrow and repentance. For no sin great or small is ever forgiven after we have come to the use of reason without sorrow and purpose of amendment.
Mass remits the punishment of the living due to mortal and venial sins after the guilt has been forgiven in virtue of its being satisfactory. This remission is infallible, relying on the merits of Christ; but to what extent punishment is remitted remains unknown. St. Thomas says: "Although this offering of the Mass, so far as its quality goes, is sufficient to cancel all the pain due to sin on this earth, nevertheless it is satisfactory to those for whom it is offered or to the offerer according to the quality of his devotion, and not for all the punishment due to his sin." (S. Th. 3. q. 79. ad 3.)
In the case of the dead, Mass infallibly cancels a portion of the punishment in Purgatory, though how much we cannot tell. The Church sanctions a perpetual Mass for the same soul, and thereby admits that she does not know how far the satisfactions of Christ are applied to that soul.
Again, it should be remembered that the propitiatory or appeasing power of the Mass saves the world in general and men in particular from many punishments which otherwise their sins would receive, such as war, famine, plague, sickness, and other temporal misfortunes.