The notion that all paths lead to the same destination is quite a new development in Hinduism. Ancient Acharyas belonging to different sects never claimed so. On the contrary, they debated a lot to prove that other schools were wrong. This intellectual exercise was central to Hindu thoughts for Millenium. For example, think about the debates between different schools of Hindu philosophy - Advaita, Samkhya, Nyaya and other Vedanta schools, etc.
However, in modern times, this intellectual vigour died. It got replaced by an over-simplistic view that all paths lead to the same destination. It has almost become a void slogan. Modern Gurus more or less actively propagates this idea or at least support it.
Ramakrishna even articulated it as one of his central teachings. He famously said there are as many ways as many beliefs ("Jato Mat Tato Path").
However, this is a blatant contradiction to the Hindu Scripture. Please note that here I am talking about Hindu scripture (Sruti - Vedas). So please do not quote from Bhagavad Gita. Bhagavad Gita is part of Mahabharata, classified as Smriti (tradition).
Here are the relevant verses from the Vedas -
"They say: It is Indra, Mitra, Varuna, and Agni, Divya (the shining one), and Garutman (the mighty universal spirit). Though the truth is one, sages describe the one being in various ways ("Ekam Sad Vipra Bahudha Vadanti"). They say it is Agni, Yama (the ordained of the world), and the Matarisvan (the life energy of the Universe) - Rgveda I.146.64
Thus, you see it clearly says that there are different names for the same destination. However, there is just one path to reach that destination.
Always performing works one should wish to live a hundred years. If you live thus as a man, there is no other way than this (Na Anyatha Asti) by which Karma does not adhere to you. - Yajurveda (Shukla) XL.2
Liberation is the freedom from the bondage of Karma. This verse clearly says that one can get rid of the karmic bondage only by performing actions (not refraining from it). So basically, it talks about Karma yoga. Here we are talking about Karma performed without attachment. Works done in this spirit do not bind the soul, rather liberates.
In this Vedic scheme, where does exactly Ramakrishna's worldview fit? Can anybody cite any Vedic verse that talks about multiple paths?