I have an alternate and, I think, unusual view on this subject, which is that Christianity actually teaches that God's existence is, by design, not proven or provable by nature at all. I think this follows from these facts:
1. Faith is the substance of things hoped for (Hebrews 11), and we don't possess [proof of] what we hope for (Romans 8:24).
* This means that anything the scripture places in the realm "of faith" is something that we don't have tangible proof of
* For example, we have righteousness "by faith" but no undeniable proof of this righteousness, as in Galatians 5:5: "For through the Spirit we eagerly await by faith the righteousness for which we hope"
2. The existence of God, and the creation of the world, are placed in the realm of faith by the scripture.
* "Without faith it is impossible to please God, because anyone who comes to him must believe that he exists" (Hebrews 11:6)
* "Through faith we understand that the worlds were framed by the word of God, so that things which are seen were not made of things which do appear." (Hebrews 11:3)
Therefore the existence of God and the creation of the world are purposefully consigned by God to the realm of unverifiable facts.
This makes sense intuitively, because it makes the gospel appear weak and foolish in accordance with 1 Corinthians 1:27, and prevents anyone from approaching the subject based on their own intellectual powers as in Matthew 11:25.
And in light of this, the verse you mentioned in Romans can be taken to be about mankind, who were originally literally shown things by God Himself corporately, rather than about any particular man or all men individually.
In my opinion, this logic precludes apologists from appealing to natural theology as a legitimate way of convincing someone to believe in God or to convert to the Christian faith. According to this line of thinking, submitting evidence to someone that can be verified by observation in order to convince someone that God exists or that Christ is God is actually undermining faith.