As the question notes, the Ramban says that the first mitzvah of the Aseres haDiberos cannot be to believe in G-d because the whole concept of "mitzvah - commandment" pre-supposes the existence of a Commander.
There are many Rishonim who do not see this as a problem. One could have a little belief and be commanded to learn enough to know, for example. Or one could just not see it as a problem to begin with - the non-believer is commanded whether or not he accepts that there is a G-d who did command him. Just as not knowing your country has a legislature doesn't stop them from passing laws that effect you.
So, there are two possibilities: either belief in G-d is indeed one of the 613 commandments, or most commandments - if not - require such belief as a precondition.
So yes, such belief is required, whether one can call it a mitzvah or whether one likes the Ramban's argument not to.
There are also the many specific mitzvos to commemorate Yetzias Mitzrayim - the Exodus, which would require believing in a G-d Who took us out of Egypt. As well as Shabbos, that the world was created (and, according to the Devarim version of the Aseres haDiberos, Shabbos is about Yetzias Mitzrayim too).
As for "blind faith", we aren't commanded to have "blind faith". Yes, the generation that lived through the Exodus and national revelation a Sinai had far more evidence to work with. Anyone who lived during the era of miracles would have more reason to believe than we do. (On the other hand, they likely only experienced miracles because they were of the sort that already believed just as well akready.)
But there is plenty of evidence around today too. That feeling one gets when totally immersed in a Shabbos that is working. How does a Shabbos of 39 Melakhos work so much better than a vague and intuitive day of rest anyway? Or the elegence of a good halachic discussion one it all finally clicks. Judaism is beautiful like an elegant math proof. And like that proof, the beauty is an assessment of something actually inherent in the ideas and how they interact and feel when we try living them.