The dictionary defines reconstruction as:
1. to construct or form again; rebuild: to reconstruct a Greek vase from fragments
2. to form a picture of (a crime, past event, etc) by piecing together evidence or acting out a version of what might have taken place
[World English Dictionary]
It can therefore be fair to say that Religious Reconstruction is hardly the act of making something exactly as it happened or was done before. That’s not to say that reconstruction is not sometimes meticulous, but it’s also not the act of making a broken vase unbroken — after all, more often than not, you can see the glue holding the fragments together, and most significantly, you can almost always see that there are fragments being held together. Another popular analogy is to take a house that has been damaged by flood or by fire and rebuild it. You’re not going to make it exactly as it was before, you know this. If the house was old enough to have lead paint in it, well, good luck making it exactly like it was before. If it was that old, then there was probably some electrical wiring and water or heat piping that would not fit with modern codes — again, good luck with that. A thing that is reconstructed is never exactly the same as it was before — if it was, then it wouldn’t need to be reconstructed, that’s just basic facts.
Furthermore, how to reconstruct, say, ancient art is often debated by art historians, art restorers, and archaeologists. A basic Google search for “reconstructed Venus de Milo [Aphrodite of Milos]” will turn up many different ideas of how the statue’s arms should be positioned and what, if anything, she’d have in her hands. Some of these ideas are obviously not supported by easily sourced evidence of how the statue was originally found and what she was found with, but some of the photo-manipulations obviously show skill and knowledge (albeit, knowledge of basic art) that these would be fair hypotheses if there was no other evidence.
Religious reconstruction is nothing more than forming a hypothesis, an educated guess, and more often than not, forming several hypotheses of what could have been, or what upgrades to the electrical wiring would need to be made. While there is a wealth of evidence in existence of what the ancient Hellenes (and other tribes) did and did not practise and believe, there are still a lot of gaps that could use some putty, a lot of corroded pipes that need replacing, and a lot of questions whose original answers have long crumbled away into dust, but those questions still need answers.
The long and short of it is that arguing over who is and is not “recon” is no less mental masturbation than any other mental exercises that have no real-world applications. It’s not as important as actual practise; if you’re actually practising your religion in a form and spirit consistent with that of ancient Hellas, then this will speak for itself. If you are not, then that is between you and the Theoi to judge whether or not it is improper or impious.