I don't know if someone read the whole story, but the reason why the DNA test issn't done is very clear.
If there's a proof that Heino Tammet IS Alexei, then he is the crown-prince.
"The accepted version of history says that sixteen months earlier Nicholas had abdicated the throne for both himself and his son. However, there is some question about the validity of the Ex-Tsar's actions in March of 1917. Nicholas abdicated not once, but twice. The first time he had abdicated for himself but on the second occasion, some six or more hours later, he wrote a second abdication document that passed the throne to his brother Michael in order to protect his ailing son. Although it has always been assumed that that document took the throne away from the Tsarevich Alexei, there are two problems with that assumption.
The second abdication document says, "Not wishing to part with our dear son, we transfer our legacy to our brother". That wording does not actually say that Nicholas was abdicating on his son's behalf. It neatly sidesteps the question of Alexei's claim to the throne and places it in Michael's lap. That could well have left Alexei open to make a renewed claim at a later date.
The more important point is that the second document was probably not legal because Nicholas had already abdicated six hours earlier when he signed his name to the first set of papers. He was no longer the Tsar when the second document was written and no longer in a legal position to abdicate on his son's behalf. When Nicholas put pen to paper for that second time Alexei was the Tsar! The son's abdication was his own decision to make... not his father's!"
-------------------------------------------------------
Sounds logical to me.....

Therefor IF Heino is Alexei, then he was the Tsar since 1918 till 1977, and that makes his oldest son the current Tsar.

And in that case he could stand up for his right for the trone and the Romanov tressuries.