No, the lessons from history suggest not.
If this is true, then a country holding an official religious stance does not inevitably lead to repression of religion and free thought. The next question that would proceed from there, then, is why such actually happens.
With said, though, I think that you're jumping to unwarranted conclusions, if you're actually trying to use the ensuing arguments. I'll deal with them more directly, though.
From the Wiki article on "Christian states":
https://en.wikipedia.org/wiki/Christian_state
Today, several nations officially identify themselves as Christian states or have state churches, including Argentina,[8] Costa Rica,[9] Denmark,[10] England,[11] Faroe Islands,[12] Greece,[13] Greenland,[14] Iceland,[15] Liechtenstein,[16] Malta,[17] Monaco,[18] Norway,[19] Samoa,[20] Tonga,[21] Tuvalu,[22] Vatican City,[23] and Zambia.[24] A Christian state stands in contrast to a secular state,[25] an atheist state,[26] or another religious state, such as a Jewish state,[27] or an Islamic state.[28]
The article notes that many Christian states have turned secular. Since this has been done without revolution in many cases, it suggests that Christian states allowed themselves to change and adopt secular values for those states.
There are a couple things that are worth touching on here. First, that article doesn't deal at all with the repression of religion or free thought in the first place, thus, it is of tangential value, at best, to what's in question there. The articles on, for example,
religious persecution would likely be of more use. For example -
The tendency of societies or groups within society to alienate or repress different subcultures is a recurrent theme in human history. Moreover, because a person's religion often determines to a significant extent his or her morality, worldview, self-image, attitudes towards others, and overall personal identity, religious differences can be significant cultural, personal, and social factors.
Religious persecution may be triggered by religious bigotry (i.e. members of a dominant group denigrating religions other than their own) or by the state when it views a particular religious group as a threat to its interests or security. At a societal level, this dehumanisation of a particular religious group may readily turn into violence or other forms of persecution. Indeed, in many countries, religious persecution has resulted in so much violence that it is considered a human rights problem.
With that said, at last check, religious tolerance and Christianity only really started becoming at all related after a number of outright wars between Protestants and Catholics. The 30 Years War, especially. Second, your reasoning here is very, very flimsy and dodges addressing the reasoning for why the states that changed to secular states did so in the first place. Third, this line of argument gives the impression that you want to quietly narrow what kinds of repression of religion and free thought that you're talking about to some of the more extreme forms and ignore the many rather common and less extreme ways that governments that have official religious views tend to support that view to the expense of other views. Fourth, it may be worth noting that atheist China actually did go out of their way to peacefully change their constitution to include provisions for religious freedom, which has helped the situation there, though more before the 6-10 Office and Xi. Thus, China would qualify as an example of an atheist state that did exactly what you're talking about, albeit under a notably more authoritarian government model than an actual democracy.
In the article under "State atheism":
https://en.wikipedia.org/wiki/State_atheism
State atheism is the incorporation of positive atheism or non-theism into political regimes, particularly associated with Soviet systems...
The Soviet Union attempted to suppress public religious expression over wide areas of its influence, including places such as central Asia. Currently, only China, Cuba, North Korea and Vietnam are officially atheist.
Bolding mine, of course. That phrase, alone, should tip you off, very clearly, that they're not dealing with any possible officially atheist government here. They're dealing with governments that were rather repressive by design. That atheism was chosen as a state religion to be enforced was fairly certainly a political move to limit or eliminate the power that religious groups (among the rest of the ideologies being repressed) could bring to bear, rather than any real ideological consensus. If we deal with China, specifically, of course, there's a very, very long history there of the leadership repressing religion and free thought that is not in line with the leadership's desires.
Since the old Soviet Union moved from communism to the new Russian government, religious freedom has increased, suggesting that moving away from state atheism results in more religious freedom.
While true, this serves as an extremely limited data point, especially given the actual nature of the old Soviet Union's "commitment" to atheism. Reasonable analogies could easily be made to a number of countries that strongly endorsed a state religion and eventually moved away from them in light of the general population becoming unhappy at the religious persecution that they were seeing.