First of all, a clarification. The Muslim rulers of India may not have abolished the caste system, they certainly played a role in weakening it. At least some people converted to Islam to escape the caste system (especially via the Sufi orders which emphasized equality and syncretism), though it's also true that upper-class converts were often viewed in more favorable terms. But the history of Islamic rule in India is long and diverse, and individual rulers varied greatly in how they approached the issue.
Now I am going to use Bengal as a case study for why Muslim rulers in India did not abolish the caste system. Bengal is an interesting case since it became a majority Muslim region despite a general lack of incentivization (such as caste abolition as mentioned by OP) or force in terms of conversion.
Nature of Muslim Conquests
The early Muslim invasions of India often featured small armies led by Muslim leaders (the soldiers themselves can be a varied lot) raiding the borderlands. If the raids were successful, they often doubled down and tried to take over the ruling apparatus. The primary motive was purely economical, they had no wish to run a religious state or an ethnostate.
Now, the size of Muslim armies (which were themselves diverse, mind you), when compared to the complex bureaucracy and the massive population subjected to them, dictated the terms of conquest. It was simply not feasible to replace a running aristocracy and rule over populous regions by putting religion first. To make wealth out of these conquests and to make them sustainable, they had to govern - and who better to govern than the existing Brahmin and upper classes who already knew the lay of the land, and were the most literate section of the society?
So the leaders of these Muslim invasions simply replaced the ruling structure with themselves at the top, but they kept everything intact. Later on, partly due to immigration and partly by conversion, the aristocracy had a higher proportion of Muslims. But merit was often more important to individual rulers than religion, so officials of diverse origins continued to be appointed. Non-muslim officials were often less threatening, something I discuss later.
The state religion was changed to Islam, but this translated to little more than different coinage. Many rulers found it advantageous to even fund temples, so as to keep the Hindu upper classes happy. This is because temples were not just places of worship, but also symbols of power and wealth. A lot of people have a misconception that Muslim raiders burnt or destroyed temples out of fanaticism, but it was actually for delegitimizing its patrons (such as the ruling dynasty) and looting the valuables in it.
The Threat of a Muslim Aristocracy
The nature of the medieval Indian state necessitated that there be an aristocracy, and a reason why the Muslim rulers did not abolish the class system is that a Muslim aristocracy offered few benefits compared to their Hindu counterparts. On the contrary, the Muslim nobility was a bigger threat to their power. Time and time again we see this pattern of Muslim dynasties being replaced by other Muslim dynasties (such as the Ghurids toppling the Ghaznavids), or Muslim nobles deposing their overlords (such as the Tughlaqs who replaced Khusrau Khan, who in turn had replaced the Khiljis after converting to Islam), or Muslim nobles declaring independence (such as Shamsuddin Ilyas Shah of Bengal), or even Muslim slaves replacing their masters (Mamluks of Delhi, or the Habshi dynasty in Bengal). Sure we can find Hindu nobles and generals doing the same ( such as Raja Ganesha in Bengal, or Hemu against the Sur dynasty), but in general, they were less frequent. Considering all of these, it was not particularly imperative for Muslim rulers to abolish the caste system- why annoy the existing Hindu aristocracy when their Muslim counterpart has such glaring weaknesses?
But as I said, the distinction between the upper strata of the Hindu caste system (Brahmins, Kshatriya, Kayastha, etc.) became fuzzier, so Muslim rule indeed had an effect in loosening up the caste system if not necessarily abolishing it.
The Psychology of Muslim Rulers
Today people may read Islamic scriptures and derive a message of equality (at least within the faith), but no religion operates in a vacuum. Culture plays just as important a role in how religion is interpreted. The aforementioned Muslim rulers came from the Persianate world (in terms of culture; ethnically their origins could be anywhere from Iran, Afghanistan, or C Asia), where class distinctions are not just necessary, but desirable. Now, this class distinction is not the exact same as a caste system (where people are born into a caste which they can't get out of and which dictates their professions), but they were not particularly motivated in preaching freedom and equality. Once again, the big exception was Sufi orders, who were alternately patronized and seen as suspicious (often both at the same time) by rulers.
It is interesting to see that by the end of Muslim rule, it was the Persianate class system that was more in vogue (think of the Muslims employed by the Marathas, or the Hindus employed by Bengal/Mysore), so in reality, it was simply one kind of social hierarchy partly changing the other.
Tl;dr
The Muslim rulers of India did not abolish the caste system because it provided no particular benefit to them.
Sources (including but not limited to)
"The rise of Islam and the Bengal frontier, 1204-1760" by Richard M. Eaton