Hardly significant. If being Orthodox increased birth rates to the replacement rate then it would have occurred already. Most Orthodox are nominal, like Catholics, who themselves have lower fertility than even Protestants in many European countries (who also have low rates).
Orthodox are not Mormons (3.4 babies in US) nor Haredi Jewish (6.6 in Israel) (Globally, Christians have a birth rate of 2.7 children per woman. But Muslims have a higher rate, namely, an average of 3.1 children per woman.).
If Putin was really determined to increase the birth rate, he would ban contraception (including by his women)
I don't think the Russian authorities are interested in the divisions within Islam, Judaism, or Buddhism. Russia was traditionally an Orthodox country, and Protestant denominations, the Roman Catholic Church, and splinter groups like the Mormons undermine the unity that Orthodoxy brought to Tsarist Russia. Additionally, all these denominations have Western ties.
Which "ties" does not mean attacking Russia unless attacked, nor support for the Western Left, but it means believing in freedom of speech and religion, which cannot be allowed under a Putin dictatorship. Thus only a state church, an an organ of the state, can be allowed. Which therefore is the real reason Putin attacks conservative evangelicals who support his laws against LGBTQ and abortion on demand.