Nope.
Paul uses the Aorist in his argument, pointing to a single event (ie. Adam's sin), not multiple events (ie. everyone sinning in and of themselves). Second, "through" is translating δια + the genitive, and is the proper translation. But, the idea of "through" is instrumental as in death happens because of sin.
Here is Leon Morris' take on it:
The aorist points to one act, the act of Adam; we would expect the present or the imperfect if the apostle were thinking of the continuing sins of all people. Paul says that all sinned in Adam, not in imitating him (cf. Bruce). And it ignores the context with its strong insistence on the sin of one man (not all of us) as the cause of the trouble.
Leon Morris, The Epistle to the Romans, The Pillar New Testament Commentary (Grand Rapids, MI; Leicester, England: W.B. Eerdmans; Inter-Varsity Press, 1988), 231–232.
What is more, Paul's argument in v. 18-19 leaves no doubt:
[18] Therefore, as one trespass led to condemnation for all men, so one act of righteousness leads to justification and life for all men. [19] For as by the one man's disobedience the many were made sinners, so by the one man's obedience the many will be made righteous. (Romans 5:18–19 ESV)
Whether one calls it "Original Sin" or something else, the historic doctrine of Original Sin is exactly what Paul is detailing here.
The Archangel