Come on, Byblos . . . I've heard what could be classified as stretching before, but this takes the cake. The theme of death runs deep through Gen 2-5. Notice:
2:17 - God warns man that if disobeys, then he will die;
3:3 - Eve repeats (with expansion) God's warning of death;
3:4: - The serpent calls God a liar, directly claiming Eve will not die;
3:19 - Adam is promised that he will die
Before going any further, note that there is no discussion of "spiritual death" anywhere in this passage. Perhaps it can be inferred from their eyes being opened to their nakedness, but even that is pressing the text, for the fulfillment of 2:17 lies far more clearly in 3:19 than in 3:7!
Now, if physical death is the actual punishment, we would expect much more discussion about such a matter, and so we see it:
3:21 - God makes garments, which implies an animal's death (the first sacrifice)
3:22 - God banishes Adam and Eve from the garden to ensure that they do not live forever (not talking spiritually here!)
4:8 - Cain kills Abel, the first human death
4:14 - Cain fears being killed
4:23 - Lamech brags over his killing two men
4:25 - Abel's murder is emphasized
Next, you have chapter 5, which repeatedly ends in that dirge, "And he died" (8 times), with only one exception, Enoch in the seventh generation. But Enoch did
not die because "Enoch walked with God." Finally, in chapter six, you have the flood coming because men were so murderous.
The point of all this is painfully obvious. Death reigns because Adam disobeyed God. Enoch and Noah stand in sharp relief to that. Death is the direct result of sin; put different, these chapters teach that death is a sin-problem. So, yes, physical death is clearly the penalty for Adam's sin.
edit:
And on this, I'll have to
agree with your church:
(1) The sin of Adam has injured the human race at least in the sense that it has introduced death -- "Wherefore as by one man sin entered into this world and by sin death; and so death passed upon all men". Here there is question of physical death. First, the literal meaning of the word ought to be presumed unless there be some reason to the contrary. Second, there is an allusion in this verse to a passage in the Book of Wisdom in which, as may be seen from the context, there is question of physical death. Wisdom 2:24: "But by the envy of the devil death came into the world". Cf. Genesis 2:17; 3:3, 19; and another parallel passage in St. Paul himself, 1 Corinthians 15:21: "For by a man came death and by a man the resurrection of the dead". Here there can be question only of physical death, since it is opposed to corporal resurrection, which is the subject of the whole chapter.
(2) Adam by his fault transmitted to us not only death but also sin, "for as by the disobedience of one man many [i.e., all men] were made sinners" (Romans 5:19). How then could the Pelagians, and at a later period Zwingli, say that St. Paul speaks only of the transmission of physical death? If according to them we must read death where the Apostle wrote sin, we should also read that the disobedience of Adam has made us mortal where the Apostle writes that it has made us sinners. But the word sinner has never meant mortal, nor has sin ever meant death. Also in verse 12, which corresponds to verse 19, we see that by one man two things have been brought on all men, sin and death, the one being the consequence of the other and therefore not identical with it.
(3) Since Adam transmits death to his children by way of generation when he begets them mortal, it is by generation also that he transmits to them sin, for the Apostle presents these two effects as produced at the same time and by the same causality. The explanation of the Pelagians differs from that of St. Paul. According to them the child who receives mortality at his birth receives sin from Adam only at a later period when he knows the sin of the first man and is inclined to imitate it. The causality of Adam as regards mortality would, therefore, be completely different from his causality as regards sin. Moreover, this supposed influence of the bad example of Adam is almost chimerical; even the faithful when they sin do not sin on account of Adam's bad example, a fortiori infidels who are completely ignorant of the history of the first man. And yet all men are, by the influence of Adam, sinners and condemned (Romans 5:18, 19). The influence of Adam cannot, therefore, be the influence of his bad example which we imitate (Augustine, "Contra julian.", VI, xxiv, 75).