Good question. i will try my level best to answer this question. i will be happy if it gives you satisfaction. Now we will go to the answer.
my answer: It is true as per history that Shri Adi Shankaracharya propagate his teachings and done works in sanskrit language. why is the question now. why he wrote and why he do propagandas ? we will see now.
The following mensioned are in sanskrit language:
'Veda' means knowledge. The Vedas are the sacred texts for Hindu rituals and for the religion itself.
RigVeda, YajurVeda, AtharvaVeda and SamaVeda are the four Vedas.
The Rig Veda comprises of 1028 hymns in praise of various gods. These hymns or 'Suktas' are arranged in 10 'Mandalas' or books. The Yajur Veda is mostly prose and makes use of the Rig Veda to explain sacrifices. The Atharva Veda is a collection of spells and charms. The Sama Veda is almost wholly drawn from the Rig Veda. All the above mentioned scriptures are part of the Vedic Literature.
While a scripture, vedas, upnishads etc.,are in a specific language, it is good to learn, to spread, to create meanings in the same language. Suppose as you said if Adi shankara done the works in any other language, the meaning while translating the mantras as well as the sound of mantras may change and all opportunities are there. Hence, Shri Adi Sankara chose the origin of vedas which were in sanskrit felt better to spread. Moreover, the sanskrit language flourished all over the south Asia on those days like Tamil language in south india.
Sanskrit Language used in support of his Thesis:
Shri Adi Shankara is believed to be the organizer of the Dashanami monastic order and the founder of the Shanmata tradition of worship. His works in Sanskrit, all of which are extant today, concern themselves with establishing the doctrine of Advaita (Nondualism). He also established the importance of monastic life as sanctioned in the Upanishads and Brahma Sutra, in a time when the Mimamsa school established strict ritualism and ridiculed monasticism.
Shri Adi Shankara relied entirely on the Upanishads for reference concerning Brahman and wrote copious commentaries on the Vedic Canon (Brahma Sutra, Principal Upanishads and Bhagavadgita) in support of his thesis. The main opponent in his work is the Mimamsa school of thought, though he also offers some arguments against the views of some other schools like Samkhya and certain schools of Buddhism that he was familiar with.
If Shri Adi Sankara done the works in other than sanskrit, it will not help the people on religious texts. so, he chose the sanskrit language. Importantly, there were more fake religions at the time of Shri Adi Sankara period. Shri Adi Sankara felt that it will be better to argue, to find the truths with those fake religions, he used sanskrit language besides the Bhuddhism religion too.
The more important:
Considering this a healthy debate, i chose to answer to clear the doubts of all. Neither Malayalam the language, nor its script, nor the Devanagari script are attested at Shri Adi Shankara's time. His ethnicity is now assumed to Malayali because of his place of birth, but given that the area could be considered to be part of the wider 'Tamil country' of those times, so Dravida naadu. The debate about the Devanagari and Malayalam scripts was quite a while ago, but I just want to clarify something. Devanagari is not, and certainly was not in the past, "the" Sanskrit script. Sanskrit can be and has historically been written in all or most Brahmi-derived scripts. A Malayali is quite likely to use the Malayalam script to write in Sanskrit. However, neither of these scripts would have been around in Shankaracharya's time.
A movie in full sanskrit: The first and only Indian movie to be made in Sanskrit. The movie follows the life and times of Sankara - the founder of the non-duality (Advaita) school of Indian philosophy.
Thanks for the opportunity !!!