So /u/holafun is downvoted to the negatives but really, his question is intertwined with yours. It's really only when "westerners" made note of it and began to criticize it that you find Muslims having moral qualms about the issue.

If you look at anti-Islamic polemics, you'll find no dearth of criticism of Muhammad and women. However, what you won't find is much criticism of his marriage to Aisha. You can look at writers like John of Damascus and Matthew of Paris and Voltaire. Plenty of harsh language and sharp attacks against Muhammad. Plenty of criticism through the ages of his marriage to Zainab bint Ja'sh. No mention of Aisha's age though. When his marriage to Aisha is attacked, it's usually attacked as a sign of his obsession with her. According to Muslim narratives, he had seen Aisha in a dream in which he was told that she would be his wife in this world and the next. That was criticized, that he was so driven by lust that he saw her in his dream and had to marry her. But her age? No. You don't see it being criticized. That's not too surprising consider that in pre-modern societies, early marriage was not abnormal.

This changes around, as far as I can tell, the early 1900's, coinciding with the decreased acceptance of child marriage. The earliest written instance I personally have seen of criticism about the marriage due to age is David Margoliouth in 1905. In Mohammed and the Rise of Islam he describes the marriage as an "ill-advised union".

Once western writers start criticizing the age, Muslims began having moral qualms as well. The issue is that Muslims consider Muhammad to the perfect man. So you have two options. Either you have to admit that it's not inherently immoral for such an age gap to exist (but that it may be wrong based on cultural context and in certain societies) or you deny the age gap in the first place. The earliest Muslim to take the second approach is, based on my research at least, Abbas al-Aqqad. Al-Aqqad is not a Muslim scholar. He was an Egyptian journalist. More anti-colonial and pro-Egyptian than an Islamist. He makes an argument using a dizzying array of indirect references and math and frank speculation to say that Aisha was around 15 at the time of the marriage. This argument is adapted by other journalists and Arabists (most of these are writers in Arabic literature departments in Egypt). It's not really something that Muslim scholars adopt because, frankly, it's a weak argument. I'm not aware of any Muslim scholar seriously believing Aisha was anything other than 9. I've been told that Taha Jabir Al-Alwani did write as much but my (admittedly brief) search for his writing on the subject came up empty. Regardless, it's at most a position that a handful of modern Muslim scholars may hold. It’s a weak historical argument, to be sure. I’ve looked at the various arguments Muslim apologists use to say she was older than 9. They all hinge on indirect math and comparing her age to her sister's age or calculating age based on subtracting events. None of them hold water in my opinion. Dr. Denise Spellberg has her own argument which is that the age was reduced to emphasize virginity. I honestly find that to be an even weaker argument.

The other approach was to say that it's not inherently immoral for such an age gap to exist but that it may be contextually immoral in certain times and places. This tends to be the argument adopted by Muslim scholars. People like Ali Tantawi and Abdel Haleem Mahmoud took great care to say that inherently there is nothing wrong with such marriages but that in the modern world, restricting marriage of young children promote social welfare. Both of these men, highly influential scholars, defended the right of the government to ban underage marriage by saying that it is the right of the government to restrict such marriages if they cause more harm than good but that Muslims can not say there is something intrinsically wrong with such marriage in all times and in all places.

Since the 20th century, you'll find that whatever of the two approaches Muslims take, they do take one. Many biographies of Muhammad written by Muslims after the 20th century will mention the age of Aisha and then explain it in one of the two ways I mentioned. Either that she was never that age or that her age was fine for that time and place but that it's not something to be repeated in the modern world. Prior to the 20th century, neither approach would have been needed as neither Muslims nor non-Muslims saw anything objectionable about the age.

There's a really good section in Jonathan Brown's Misquoting Muhammad which talks about how Muslims have grappled with Aisha's age since the 20th century. I'd recommend that as a starting point. I honestly don't know of any full books on the subject. Plenty of polemics in either direction but no academic study of the polemics themselves that I am aware of.

Also when did westerners start to make note of this? And use it to criticize the the religion?