The chief linguistic difference between the rites was that [Roman] Catholic language was, precisely, deliberately unambiguous and Anglican language (because the same Eucharistic prayer had to gain acceptance from Anglo-Catholics and evangelicals alike) was inevitably ambiguous.
What Oddie judges as the weakness of ambiguity, Anglicans can view as a gift to the universal church - acknowledging both the catholic recognition of the eucharistic sacrifice and the Reformation insistence on the uniqueness of Christ's atoning death on the cross. Thus the BCP's prayer of consecration memorably proclaims the offering of Christ once made:
Almighty God, our heavenly Father, who of thy tender mercy didst give thine only Son Jesus Christ to suffer death upon the Cross for our redemption; who made there (by his one oblation of himself once offered) a full, perfect, and sufficient sacrifice, oblation, and satisfaction, for the sins of the whole world.
The very next words, however, state that the eucharist makes the sacrifice of Christ present now:
and did institute, and in his holy Gospel command us to continue, a perpetual memory of that his precious death, until his coming again.
The post-communion prayer of oblation then recognises the explicitly sacrifical nature of the eucharist:
O Lord and heavenly Father, we thy humble servants entirely desire thy fatherly goodness mercifully to accept this our sacrifice of praise and thanksgiving; most humbly beseeching thee to grant, that by the merits and death of thy Son Jesus Christ, and through faith in his blood, we and all thy whole Church may obtain remission of our sins, and all other benefits of his passion.
The same rhythm is to be found in contemporary Anglican liturgies. For example, the Church of Ireland's eucharistic prayer I gives thanks for the sacrifice of Christ on the cross:
he made there the one complete and all-sufficient sacrifice for the sins of the whole world.
The prayer continues after the narrative of the institution:
Accept through him, our great high priest, this our sacrifice of praise and thanksgiving.
The wording quite deliberately connects the language of eucharistic sacrifice with the ongoing priesthood of Christ.
This rhythm, of acknowledging both the offering of Christ once made and the eucharist of making his sacrifice present in the now, is not ambiguity. It is holding together key affirmations of Scripture - that the offering of Christ was made once upon the cross and that the sacrament of the eucharist makes Christ's sacrifice present in the Church. To deny the former obscures the uniqueness of the Cross. To deny the latter fails to recognise (in the words of Article XXVIII) that the eucharist "is a Sacrament of our Redemption by Christ's death".
It is perhaps in Saepius Officio - the response of Canterbury and York to Leo XIII's Apostolica Curae - that we find most clearly summarised how Anglican liturgy and doctrine seeks to recognise both of these key affirmations of Scripture:
We truly teach the doctrine of Eucharistic sacrifice and do not believe it to be a “nude commemoration of the Sacrifice of the Cross,” an opinion which seems to be attributed to us by the quotation made from that Council. But we think it sufficient in the Liturgy which we use in celebrating the holy Eucharist,—while lifting up our hearts to the Lord, and when now consecrating the gifts already offered that they may become to us the Body and Blood of our Lord Jesus Christ,—to signify the sacrifice which is offered at that point of the service in such terms as these. We continue a perpetual memory of the precious death of Christ, who is our Advocate with the Father and the propitiation for our sins, according to His precept, until His coming again. For first we offer the sacrifice of praise and thanksgiving; then next we plead and represent before the Father the sacrifice of the cross, and by it we confidently entreat remission of sins and all other benefits of the Lord’s Passion for all the whole Church; and lastly we offer the sacrifice of ourselves to the Creator of all things which we have already signified by the oblations of His creatures. This whole action, in which the people has necessarily to take its part with the Priest, we are accustomed to call the Eucharistic sacrifice.
We do teach the doctrine of eucharistic sacrifice, but we do so mindful of the necessary Reformation protest against separating the eucharist from the one oblation of Christ finished upon the Cross. This is not ambiguity. It is rejoicing in the catholic tradition and respecting Reformation insights.