I am trying to check if the story of Saint Helena mother of Constantine 1 finding the true cross logically follows or not.

The most common story goes ( Wikipedia:Church of the Holy Sepulchre)

Following the siege of Jerusalem in AD 70 during the First Jewish–Roman War, Jerusalem had been reduced to ruins. In AD 130, the Roman emperor Hadrian began the building of a Roman colony, the new city of Aelia Capitolina, on the site. Circa AD 135, he ordered that a cave containing a rock-cut tomb[d] be filled in to create a flat foundation for a temple dedicated to Jupiter or Venus. The temple remained until the early 4th century.

After allegedly seeing a vision of a cross in the sky in 312, Constantine the Great began to favor Christianity, signed the Edict of Milan legalising the religion, and sent his mother, Helena, to Jerusalem to look for Christ's tomb. With the help of Bishop of Caesarea Eusebius and Bishop of Jerusalem Macarius, three crosses were found near a tomb; one which allegedly cured people of death was presumed to be the True Cross Jesus was crucified on, leading the Romans to believe that they had found Calvary. Constantine ordered in about 326 that the temple to Jupiter/Venus be replaced by a church. After the temple was torn down and its ruins removed, the soil was removed from the cave, revealing a rock-cut tomb that Helena and Macarius identified as the burial site of Jesus.

If crosses were re-used then this part would absolutely mean nothing "three crosses were found near a tomb" ... like they just won't leave the same three crosses laying the same exact place especially after the absolutely mess that happened in AD 70

In addition that for wood to decompose upwards of 50-100 years if left whole. This is why for composting it's best to chip the wood and she is supposed to have found remains of the thing after 4.4x the decomposition time!

I find it really oddly specific that they found the real tomb inside a pagan temple, like why of all places did the Roman emperor Hadrian pick Jesus's tomb to build a pagan temp on?

John 19:41-42 says:

41 At the place where Jesus was crucified, there was a garden, and in the garden a new tomb, in which no one had ever been laid. 42 Because it was the Jewish day of Preparation and since the tomb was nearby, they laid Jesus there.

while it's said

Helena, later known as Flavia Julia Helena Augusta, mother of Constantine the Great, was credited after her death with having discovered the fragments of the Cross and the tomb in which Jesus was buried at Golgotha. Brown University.EDU

does the verse contradict the location or?

in addition that I read this reasoning

it is far too close to the supposed site of the crucifixion. A rich and influential person like Joseph of Arimathea would never have chosen a site intended for his own burial so close to where the Romans executed criminals.

I need verified answers since I am using this for a research project.

(what matters now is the cross reuse part)

Edit : After doing more research i found that the story of helena is missing from the earliest source and not mentioned up until a decade later.

Eusebius of Caesarea at the book of Life of Constantine 337-340 AD is the earliest source about the construction of the church. Eusebius only mentions Constantine and Macarius being involved. The story goes like this

The bishop of Jerusalem Macarius asked permission from Constantine to start an excavation to search for the tomb, after demolishing a template to Aphrodite built over the area the excavators soon made a discovery

Eusebius writes

As soon as the original surface of the ground, beneath the covering of earth, appeared, immediately and contrary to all expectation, the venerable and hallowed monument of our Savior’s resurrection was discovered.

Source : life of Constantine(book iii p.133) , while he did mention Helena elsewhere in the text but he only gives her credit for sponsoring two churches

1. church of the nativity in Bethlehem
2. chapel of the ascension, on the mount of olives ,Jerusalem.

Eusebius lavishes her with praise so if she were to have been involved with Excavating the tomb of Christ, we would expect him to mention her as well, but that isn't mentioned until decades later by other Christian writers. ex. Socrates Scholasticus wrote about a century after Eusebius in his book The Ecclesiastical History which was finished in 439. also the church was supposedly already being built by the time Helena arrived in Jerusalem around 326-327(life of Constantine book iii 274 commentary) they don't say in what year/month she did actually arrive they just say "undertook a trip to Palestine". Most References say that in Beginning in 325 or 326 CE Emperor Constantine commissioned his architect, Zenobius, to build a large church in a basilica style within Roman Jerusalem So The Dates go like the following

1. siege of Jerusalem in AD 70,In AD 130, the Roman emperor Hadrian began the building of a Roman colony, Circa AD 135, he ordered that a cave containing a rock-cut tomb be filled in to create a flat foundation for a temple dedicated to Jupiter or Venus
2. 325/326 CE The Previous temple was found and after it was torn down by orders of Constantine the tomp of Christ was found underneath it and the church was ordered to be built
3. 326/327 CE Saint Helena undertook a trip to Palestine
4. 13 September 335 CE The church was finished and consecrated

The Date mixing rules the story of Helena even more. I think if we were to know the original date of Feast of the Cross. it would help pin dates down even more.

A Reference That mentions most sources about the story of the finding of the tomb