So this question covers a lot of ground, and I don't think I'm qualified to answer certain parts of it, but I can at least provide context a out the use of Arabic by Islamic practitioners around the world.

Just reiterating the correction here from the previous comments in this thread, that the actual question should be

Why didn't Arabic become ...

The first thing to talk about would probably be that the Arabic that is involved in Islamic practices is specifically Fus'ha Arabic or Classical Arabic (The English terminology might not be official, I never really had to refer to it in this Language). Its a preserved version with strict rules about the words, pronunciations, etc. This is the version that is present in the Qur'an and the various sayings attributed to the Prophet (PBUH) . To put this in perspective with an example this would be akin the likes of Shakespearean English or an earlier version compared to the vernacular Arabic spoken today which has dialects galore where there are differences in sentence structures, words, and even pronunciations of letters. For example in classical Arabic the letter (ج) can be transliterated approximately as jeem While in today's Egyptian (Misri) Dialect is more closer to geem with a hard G as in the word "Gate"

Now that we established the difference between modern day vernacular Arabic and the kind of Arabic that would be common amongst active practitioners of Islam, we can move onto the main meat of the Question. The answer is that it is in fact the lingua franca but only for a very specific subset of people, namely Ulema or Religious Scholars. The average Muslim can and often does learn to read Quranic Arabic or even just memorize the passages they need to perform their daily prayers, and they'll get by just fine. The emphasis is on read because one doesn't necessarily need to know Arabic or what the words mean to be able to pronounce the words or recite them. Other than Religious Scholars, your average everyday Ahmad likely just knows enough to read the words on the pages of the Quran - he might recognize a word or two here, have a word in his own language that matches a word he read, or read a translation of a passage after reciting the words themselves without knowing what each individual word means. Fus'ha Arabic is a language that's got it's rules and generally doesn't deviate from them except in a handful of specific circumstances that are notated on each copy of the Quran so after knowing the pronunciations of the letters and with a little practice, nearly anyone can recite it to the point that kids will start learning very young but to actually understand this secondary language is something most only dabble in.

On the other hand, we have the Ulema. Most serious scholarship is completely in Fus'ha Arabic and the general notion is that to truly be able to excel in that field, One cannot rely on translations of material as a crutch for long there. So you'll have a common language between a religious scholar sitting in Sultanahmet in Istanbul to an instructor at Madinah University to the Imam of a small local Mosque in New York for example. While there are corpuses of scholarly work in other languages such as Urdu, the importance of understanding Arabic can not be understated in that field.

That's all the context I feel qualified to offer, as to the why English/France becoming the lingua franca between Islamic countries >!I suspect Imperialism to be the answer there, But I'll leave that to someone else to provide an answer for - It would be pure speculation coming from me !<