So I did some writing on this as part of my degree. As basic chronology, yes it is basically the case that Sunni Islam became more "orthodox" after the siege of Baghdad, but to apply causation would be a serious case of post hoc ergo propter hoc.

One of the primary differences between Islam in, say, the 10th century and the 14th is, as you say, that it is much more orthodox. But this might not mean quite what you think. It's not more serious or fundamentalist or whatever. (NB: the terminology used here doesn't help at all. What I'm about to describe is often termed the "Sunni Revival" which brings up images of Christian religious revivals, put that out of your mind, it doesn't mean that, so it's a bit of a loaded term.) The basic issue is that before the 12th century the two greatest debates in Sunni Islam were between the four schools of Islamic law and as a separate matter the arguments over Islamic theology which are often divided into the two camps of Asharite and Mutazalite.

Sunni Islam became more "orthodox" in the sense that all of these issues were put aside or settled. The four madhabs stopped viewing each other as heterodox and started accepting each other as part of orthodox sunnism without question. The theological debate was eventually resolved in favor of Ghazali.

But to bring it back to your question did this happen because of the Mongols? As an intellectual concern, no. The mutual acceptance of the four madhabs as full orthodoxy goes back at least to Nizam al-Mulk. Ghazali, who is often credited with settling the theological dispute died more than a century before the sack of Baghdad. The Sunni Revival and Sunni "Internationalism" are usually attributed to the 11th or 12th century, not the 13th or 14th.

Other historians, like George Makdisi, would point out that there is less change than there is continuity, that there was a "traditionalist triumph" going back to the rise of Hanbalism and the Qadiri Creed in the 10th century or earlier.

So as causation for these intellectuals developments the mongols are out.

But that's not really the end of it, as even going into the 13th century much of this might not have been viewed as being absolutely settled. That does happen in the chaos after the sack of Baghdad as the Islamic world is forced to completely reorient itself on more solid footing, and for the first time, under the rule of non-Muslims in the heart of Islamdom.

Source wise: a good concise comparison would be between George Makdisi's essay "The Sunni Revival" and Massimo Campini's "In Defence of Sunnism: Al-Ghazali and the Seljuqs."