There is a reason to add "in Adam" since the "the wages of sin is death", both physical and spiritual, and death entered by one man, therefore death entered by Adam.
The issue in Rom 5:12 is how the death got from that one man to us, which Paul tells us is the same way death came to Adam (because he sinned) and death comes to us the same way because we all sin too. The wage of sin is death doesn't help, if we simply had death passed down to us because of Adam's sin then death wouldn't wages but an inheritance. Paul was talking about the results of
our sin when he said that.
Rom 6:20
When you were slaves of sin, you were free in regard to righteousness.
21 But what fruit were you getting at that time from the things of which you are now ashamed? The end of those things is death.
22 But now that you have been set free from sin and have become slaves of God, the fruit you get leads to sanctification and its end, eternal life.
23 For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord.
Verses 14, 17,18, and 19 all point to Adam,
Because Paul is using Adam as an illustration of Christ and the cross, but that doesn't mean we can make up our own explanation of what happened with Adam's sin and ignore what Paul actually tell us.
so I believe he needs to be object of "all have sinned". Also I believe since the phrase "all have sinned" points to the past and not the present or future, it points to Adam's sin.
I don't think the past tense helps you because Paul is quite happy to describe the entire human race and say we have all sinned Rom 3:23
all have sinned and fall short of the glory of God. This is of course speaking generally about the human race, when we look at the individual as Paul tell us about himself, as children we were once alive before we understood what sin is, but when we understood what sin was and the difference between right and wrong we all sinned. At the same time Paul describes the death that that comes from sin as a present tense ongoing process, 1Cor 15:22
For as in Adam all die. This sinning and dying spiritually in our sin is an ongoing process.
Worth pointing where the idea came from that we all sinned in Adam. It is from Romans 5:12, but Augustine's Latin translation, which instead of saying
...death spread to all men because all sinned, was translated
in quo omnes peccaverunt, '
in whom (or in which)
all sinned', and the 'in whom' was taken as Adam. And so the idea took a deep root in the church and in our understanding of sin. By the time we left the Latin bible and got back to the Greek, and realised the Romans 2:12 doesn't say in whom all sinned, the idea had a deep root in our theology and our understanding of Paul, and commentator just keep reading the idea into passages that say nothing about us sinning in Adam. For example Gill takes Rom 3:23
All have sinned and fall short of the glory of God and says "All have sinned in Adam". PNT, The People's New Testament, takes Romans 5:12 and says "For that all have sinned: The personal sins of responsible persons are not now spoken of, but all the race sinned in Adam..." But such a major claim really needs to be backed up by a clear statement that we
have all sinned in Adam and not simply read into back into texts that don't say anything of the sort.
I also believe in the "total depravity of man" and the fact that man cannot not sin. This I believe is the "sin nature", it is man's natural behavior to sin until his dead spirit is regenerated by the Holy Spirit (born again).
And the bible does teach clearly as we have seen that all have sinned and fall short of the glory of God. However I don't think it helps our understanding of this to read explanations back into scripture rather than basing or understanding on what scripture actually says. Or at least if we do, we need to recognise these ideas are speculative and keep then separate in our minds from what the text actually say.