Perhaps in a sense there is a higher fidelity, though that is at least a product of survivorship bias given the penchant for destruction of variants. But as far as I am aware, there isn't as open of a process for critical scholarship regarding the manuscripts that do exist. Also, there is the issue of what constitutes a variant given the dialect variance that exists within the manuscripts and the tolerance for at least seven distinct Arabic dialects or "readings". And in addition to the issues you mentioned, there is the unity of the text that is also in play given the relatively late collection into a single codex for the Bible compared to the Qu'ran. My minor quibble with what you had said was simply because it appeared to express a confidence in the Qu'ranic texts that the comparitive interest in critical scholarship creates a slight bias. There is also the issue of the oldest extant manuscript displaying a massive amount of variance from the later standardized texts from chapter order, number, and variant readings beyond the recognized "acceptable" variations.
Yes, I get that ... I think it's fair to say that from Uthman (650 CE) until now the Quranic textual stability is indeed higher than that of the NT writings over that time-frame; yet it's also true the earliest Quran manuscripts we have definitely show variance going back to pre-Uthmanic era. And those variants sometimes indeed changed the meaning of a verse.
The small size of the Quran, the very early (within 20 years of Mohammed) standardisation and the tradition of memorising/reciting the Quran (facilitated by its smaller size), all helped to maintain that textual stability.
It's true Quranic textual criticism is underdeveloped relatively to the Jewish/Christian ones. And indeed in the Islamic world it's common to exaggerate the Quranic stability and claim perfect preservation (which those earliest manuscripts show is not true). But in Christian traditional circles you will find the same exaggerated claim and even plain rejection of any textual criticism.
NT variants sometimes are plain copying errors, sometimes obvious 'corrections/additions' to facilitate the theology/understanding of the copyist. So all together I still would maintain the textual stability (from original author to now) of the NT is worse than that of the Quran, but that does not mean the Quranic 'revelation' is more believable or true that the NT. I believe Yeshua is the Son of God (YHWH) who died for our sins - something the Quran explicitly denies.
For mission and personal witness to Muslims - even as friends - I need to know the Quran, otherwise I can't make any comparative claims about it relative to the Bible. I need to know about the life of Mohammed in order to make comparative claims relative to Yeshua.
Be blessed!