There are many explanations given for one to become jagadguru. Some say that by writing commentaries on prasthAntrayI, one becomes a jagadguru, some say by touring India and winning debates make one jagadguru, while some say the one who spreads nAma sankirtan, the yuga dharma, is a jagadguru. Lets examine what makes one a jagadguru.
bhArat is soaked in spirituality. There are many traditions, Saiva, Sakta, vaiSnav, yOga, nyAya, cArvAka, Agama, vAmcAra mArg (left hand path), tantra, etc. Our scriptures are monotheists, but they have pluralistic approach in a way that many forms of God are given, who are manifestation of brahman. A jagadguru in my opinion is the one who teaches all paths, harmonizes all paths, do not contradict veda-s, smriti-s and purANa-s and blends all into one harmonious system. A jagadguru removes misconceptions, corrects flaws in practice like removing weed gathered over a period of time, refines it, and offers in pure distilled form.
In this context, a jagadguru must have mastery over not only one siddhAnta, but many siddhanta-s. Obviously, it is almost impossible for human being to have this knowledge. Hence only an avatar can be a jagadguru. Our AcArya Adi Sankara possessed those qualities of a jagadguru. From his biographies, and his extant works, we come to know that
All smArta-s were advaitins or all advaitins earlier were smArta-s. smArta-s are the ones who follow smriti-s and worship pandevatA (gaNeSa, Siva, Sakti, viShNu and sUrya). There are 18 smriti-s, some of them are Manu Smriti, Apastamba Dharma Smriti, Gautam, Yagnavalkya, Atri, Daksha, Shankha, etc. Smriti-s are typically called as 'notes from memory'. Great Rishi-s what had mastered Veda-s wrote smriti-s in accordance with Veda-s (shturi-s). Hence even shruti-s are also important to smArta-s. Smriti-s are also called as dharma Shastra-s (laws code, code of conduct) and are created for universal well being and harmony. Adi SankarAcArya has quoted many smriti-s in his bhashya-s. Which means that he himself gave importance to them. dharma SAstra-s, which provide moral and ethical laws, permit us to take sword for self protection and also allow Brahmins who are starving to take up occupations like trading and even take up the dharma of sword as a last resort.
If we stick to 'Brahma Satya Jagat Mithy'a, then dharma shastra-s are also useless and everything is illusion which is not true, as evident from his commentaries.
Our AchArya says, until you attain inner purity, you will have to do nitya karma and carry out your duties.
The contradictory compositions praising any form of God, or praising kruSNa / govindA or Siva or tripurAsundari were not composed because the philosophy of advaita is incomplete in itself, as claimed by some rivals. karma, upAsanA, yOga, Sakta, Agama-s etc are the foundation on which advaita rests. Advaita is the mountain peak, while ways to reach them are many. Base of mountain is very broad, but peak is only one.
There are famous SubhASitA-s which sum-up
Ruchinam vaichitryad rijukutil nana path jusham;
nrinam ekogamyastvamasi pyasamarnavmiti"
"Due to the differences in individual dispositions, people follow different paths, but you are the only destination of all of them, just as the sea is the destination of all the waters"
Akashat patitam toyam sagaram prati gacchati,
sarva deva namaskaram Keshavam prati gacchati"
"All the water fallen from the sky goes to the sea,
salutations to all the gods reaches to the KeSava"
Note: Here keSava is not kruShNa, a person, but the supreme Self, formless Brahman under it's sway trinity works. (scroll at bottom of article for explanation)
There is no shortcoming or incompleteness in his teachings, nor advaita is an imcomplete philosophy. It is the crest jewel of vedAnta, and represents highest philosophical truth after knowing this truth, nothing more needs to be known proclaims shruti.
Adi Shankaracharya only established advaita as highest philosophical truth. AcArya did not demean any form of God, nor did condemn or opposed any philosophy. Our AcArya accepted them upto a certain point, found them incomplete and asked us to rise above them. According to our AcArya, without knowledge (direct experience of Self), liberation is not possible.
He himself did not gave Jnana to any grihastha and gave utmost importance to sanyAsa for attaining Brahma-vidya, as one can fully devote all 24 hours to God. To practice advaita vedAnta, vairAgya and burning desire for liberation are extremely important. He had only 4 disciples, that too after walking the length and breadth of India. This shows that he did not teach advaita to all.
For masses, he also composed hymns for various deities , repaired temples, reconsecrated sri yantra-s. This means he encouraged deity worship, shakta-s tantra (in pure form), karma kand (mimAmsA), but he did not consider them as the supreme goal. So he asks us to rise above them after you attain inner purity.
Unfortunately some parts of his biography, as some say, is corrupted and hence rejected by rivals, and only his prasthAntrayI Bhashya is considered as authentic and are undoubtedly attributed to Adi Shankaracharya. If we blend even some of his hymns, his biography and his bhashya-s, we can understand why he is called as Jagat Guru.
According to mAdhaviya shankara Digvijay (written by VidyaraNya Svami), his gurU gOvindpAdaa is considered as incarnation of Shesha Naaga and the same shesha Naaga in earlier incarnation was Maharshi Patanjali.
saundarya lahiri is also considered as very powerful composition on Devi bhavAnI, specially the first 40 verses (out of total 100) are said to be very potent. Then there is Dakshinamurti stotra, Shiva Panchakshara stotra and Veda Sara Shiva Stotra is also attributed to him, all considered authentic atleast by Kanchi Paramacharya. They show all harmonious attitude of our AcArya.
Citing 12 purANa-s which includes 3 up-purANa-s in his Vishnu Sahasranam Bhashya shows that he was in full support of purANa-s and he did not devised symbolic meaning. Hence he even propagated bhakti. He also cited tamasic and rajasic puranas which means that unlike Vaishnava-s he did not considered them as leading to hell, or in other words, those verses are interpolations.
In his lesser known work, prabOdha sudhAkara, AcArya goes on to praise bhakti towards kruShNa as the ultimate goal of life. While describing how to meditate on various forms of bhagavAn kruShNa, AcArya goes into ecstasy and claims that for the ones socked in pure bhakti, there is no need to liberation. (pra. su 240, 250). Adi Sankara also recalls many lilA-s of kruShNa bhagavAn.
bhagavAn viShNu can be said to be our AcArya's ISTa devatA, but he never denigrated any other form of God like Siva or Sakti.
Our AcArya blended yOga, tantra (Sakta), karma kANDa, upAsanA kAnDa, GYAna kAnDa, Agama SAstra into one harmonious system and established advaita as the final destination by reconciling contradictions and difference among various religio-philosophical systems.
In my opinion, these are the reasons why our AchArya is reverentially called as Jagadguru.
Adi Shankara in his BhASya explains the meaning of the word KeSava in three different ways. The second definition is:
va = That which one has in one's possession, what is under one's sway. i.e. One who contains in himself - Lord of Creation, Preservation and Dissolution is keSava.
paramAcArya also says that in Veda-s and PurANa-s brahma and viShNu are referred to as such (ka & a) at many places.